1.
Introduction to quantum mechanics
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Quantum mechanics is the science of the very small. It explains the behaviour of matter and its interactions with energy on the scale of atoms, by contrast, classical physics only explains matter and energy on a scale familiar to human experience, including the behaviour of astronomical bodies such as the Moon. Classical physics is still used in much of science and technology. However, towards the end of the 19th century, scientists discovered phenomena in both the large and the worlds that classical physics could not explain. This article describes how physicists discovered the limitations of classical physics and these concepts are described in roughly the order in which they were first discovered. For a more complete history of the subject, see History of quantum mechanics, Light behaves in some respects like particles and in other respects like waves. Matter—particles such as electrons and atoms—exhibits wavelike behaviour too, some light sources, including neon lights, give off only certain frequencies of light. Quantum mechanics shows that light, along all other forms of electromagnetic radiation, comes in discrete units, called photons, and predicts its energies, colours. Since one never observes half a photon, a photon is a quantum, or smallest observable amount. More broadly, quantum mechanics shows that many quantities, such as angular momentum, angular momentum is required to take on one of a set of discrete allowable values, and since the gap between these values is so minute, the discontinuity is only apparent at the atomic level. Many aspects of mechanics are counterintuitive and can seem paradoxical. In the words of quantum physicist Richard Feynman, quantum mechanics deals with nature as She is – absurd, thermal radiation is electromagnetic radiation emitted from the surface of an object due to the objects internal energy. If an object is heated sufficiently, it starts to light at the red end of the spectrum. Heating it further causes the colour to change from red to yellow, white, a perfect emitter is also a perfect absorber, when it is cold, such an object looks perfectly black, because it absorbs all the light that falls on it and emits none. Consequently, a thermal emitter is known as a black body. In the late 19th century, thermal radiation had been well characterized experimentally. However, classical physics led to the Rayleigh-Jeans law, which, as shown in the figure, agrees with experimental results well at low frequencies, physicists searched for a single theory that explained all the experimental results. The first model that was able to explain the full spectrum of radiation was put forward by Max Planck in 1900
2.
Quantum mechanics
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Quantum mechanics, including quantum field theory, is a branch of physics which is the fundamental theory of nature at small scales and low energies of atoms and subatomic particles. Classical physics, the physics existing before quantum mechanics, derives from quantum mechanics as an approximation valid only at large scales, early quantum theory was profoundly reconceived in the mid-1920s. The reconceived theory is formulated in various specially developed mathematical formalisms, in one of them, a mathematical function, the wave function, provides information about the probability amplitude of position, momentum, and other physical properties of a particle. In 1803, Thomas Young, an English polymath, performed the famous experiment that he later described in a paper titled On the nature of light. This experiment played a role in the general acceptance of the wave theory of light. In 1838, Michael Faraday discovered cathode rays, Plancks hypothesis that energy is radiated and absorbed in discrete quanta precisely matched the observed patterns of black-body radiation. In 1896, Wilhelm Wien empirically determined a distribution law of black-body radiation, ludwig Boltzmann independently arrived at this result by considerations of Maxwells equations. However, it was only at high frequencies and underestimated the radiance at low frequencies. Later, Planck corrected this model using Boltzmanns statistical interpretation of thermodynamics and proposed what is now called Plancks law, following Max Plancks solution in 1900 to the black-body radiation problem, Albert Einstein offered a quantum-based theory to explain the photoelectric effect. Among the first to study quantum phenomena in nature were Arthur Compton, C. V. Raman, robert Andrews Millikan studied the photoelectric effect experimentally, and Albert Einstein developed a theory for it. In 1913, Peter Debye extended Niels Bohrs theory of structure, introducing elliptical orbits. This phase is known as old quantum theory, according to Planck, each energy element is proportional to its frequency, E = h ν, where h is Plancks constant. Planck cautiously insisted that this was simply an aspect of the processes of absorption and emission of radiation and had nothing to do with the reality of the radiation itself. In fact, he considered his quantum hypothesis a mathematical trick to get the right rather than a sizable discovery. He won the 1921 Nobel Prize in Physics for this work, Einstein further developed this idea to show that an electromagnetic wave such as light could also be described as a particle, with a discrete quantum of energy that was dependent on its frequency. The Copenhagen interpretation of Niels Bohr became widely accepted, in the mid-1920s, developments in quantum mechanics led to its becoming the standard formulation for atomic physics. In the summer of 1925, Bohr and Heisenberg published results that closed the old quantum theory, out of deference to their particle-like behavior in certain processes and measurements, light quanta came to be called photons. From Einsteins simple postulation was born a flurry of debating, theorizing, thus, the entire field of quantum physics emerged, leading to its wider acceptance at the Fifth Solvay Conference in 1927
3.
History of quantum mechanics
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The history of quantum mechanics is a fundamental part of the history of modern physics. In the years to follow, this theoretical basis slowly began to be applied to chemical structure, reactivity, Ludwig Boltzmann suggested in 1877 that the energy levels of a physical system, such as a molecule, could be discrete. He was a founder of the Austrian Mathematical Society, together with the mathematicians Gustav von Escherich, the earlier Wien approximation may be derived from Plancks law by assuming h ν ≫ k T. This statement has been called the most revolutionary sentence written by a physicist of the twentieth century and these energy quanta later came to be called photons, a term introduced by Gilbert N. Lewis in 1926. In 1913, Bohr explained the lines of the hydrogen atom, again by using quantization, in his paper of July 1913 On the Constitution of Atoms. They are collectively known as the old quantum theory, the phrase quantum physics was first used in Johnstons Plancks Universe in Light of Modern Physics. In 1923, the French physicist Louis de Broglie put forward his theory of waves by stating that particles can exhibit wave characteristics. This theory was for a particle and derived from special relativity theory. Schrödinger subsequently showed that the two approaches were equivalent, heisenberg formulated his uncertainty principle in 1927, and the Copenhagen interpretation started to take shape at about the same time. Starting around 1927, Paul Dirac began the process of unifying quantum mechanics with special relativity by proposing the Dirac equation for the electron, the Dirac equation achieves the relativistic description of the wavefunction of an electron that Schrödinger failed to obtain. It predicts electron spin and led Dirac to predict the existence of the positron and he also pioneered the use of operator theory, including the influential bra–ket notation, as described in his famous 1930 textbook. These, like other works from the founding period, still stand. The field of chemistry was pioneered by physicists Walter Heitler and Fritz London. Beginning in 1927, researchers made attempts at applying quantum mechanics to fields instead of single particles, early workers in this area include P. A. M. Dirac, W. Pauli, V. Weisskopf, and P. Jordan and this area of research culminated in the formulation of quantum electrodynamics by R. P. Feynman, F. Dyson, J. Schwinger, and S. I. Tomonaga during the 1940s. Quantum electrodynamics describes a quantum theory of electrons, positrons, and the electromagnetic field, the theory of quantum chromodynamics was formulated beginning in the early 1960s. The theory as we know it today was formulated by Politzer, Gross, thomas Youngs double-slit experiment demonstrating the wave nature of light. J. J. Thomsons cathode ray tube experiments, the study of black-body radiation between 1850 and 1900, which could not be explained without quantum concepts
4.
Classical mechanics
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In physics, classical mechanics is one of the two major sub-fields of mechanics, along with quantum mechanics. Classical mechanics is concerned with the set of physical laws describing the motion of bodies under the influence of a system of forces. The study of the motion of bodies is an ancient one, making classical mechanics one of the oldest and largest subjects in science, engineering and technology. Classical mechanics describes the motion of objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars. Within classical mechanics are fields of study that describe the behavior of solids, liquids and gases, Classical mechanics also provides extremely accurate results as long as the domain of study is restricted to large objects and the speeds involved do not approach the speed of light. When both quantum and classical mechanics cannot apply, such as at the level with high speeds. Since these aspects of physics were developed long before the emergence of quantum physics and relativity, however, a number of modern sources do include relativistic mechanics, which in their view represents classical mechanics in its most developed and accurate form. Later, more abstract and general methods were developed, leading to reformulations of classical mechanics known as Lagrangian mechanics and these advances were largely made in the 18th and 19th centuries, and they extend substantially beyond Newtons work, particularly through their use of analytical mechanics. The following introduces the concepts of classical mechanics. For simplicity, it often models real-world objects as point particles, the motion of a point particle is characterized by a small number of parameters, its position, mass, and the forces applied to it. Each of these parameters is discussed in turn, in reality, the kind of objects that classical mechanics can describe always have a non-zero size. Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the degrees of freedom. However, the results for point particles can be used to such objects by treating them as composite objects. The center of mass of a composite object behaves like a point particle, Classical mechanics uses common-sense notions of how matter and forces exist and interact. It assumes that matter and energy have definite, knowable attributes such as where an object is in space, non-relativistic mechanics also assumes that forces act instantaneously. The position of a point particle is defined with respect to a fixed reference point in space called the origin O, in space. A simple coordinate system might describe the position of a point P by means of a designated as r. In general, the point particle need not be stationary relative to O, such that r is a function of t, the time
5.
Interference (wave propagation)
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In physics, interference is a phenomenon in which two waves superpose to form a resultant wave of greater, lower, or the same amplitude. Interference effects can be observed with all types of waves, for example, light, radio, acoustic, surface water waves or matter waves. If a crest of a wave meets a crest of wave of the same frequency at the same point. If a crest of one wave meets a trough of another wave, constructive interference occurs when the phase difference between the waves is an even multiple of π, whereas destructive interference occurs when the difference is an odd multiple of π. If the difference between the phases is intermediate between two extremes, then the magnitude of the displacement of the summed waves lies between the minimum and maximum values. Consider, for example, what happens when two identical stones are dropped into a pool of water at different locations. Each stone generates a circular wave propagating outwards from the point where the stone was dropped, when the two waves overlap, the net displacement at a particular point is the sum of the displacements of the individual waves. At some points, these will be in phase, and will produce a maximum displacement, in other places, the waves will be in anti-phase, and there will be no net displacement at these points. Thus, parts of the surface will be stationary—these are seen in the figure above, prime examples of light interference are the famous Double-slit experiment, laser speckle, optical thin layers and films and interferometers. Dark areas in the slit are not available to the photons. Thin films also behave in a quantum manner, the above can be demonstrated in one dimension by deriving the formula for the sum of two waves. Suppose a second wave of the frequency and amplitude but with a different phase is also traveling to the right W2 = A cos where ϕ is the phase difference between the waves in radians. Constructive interference, If the phase difference is a multiple of pi. Interference is essentially a redistribution process. The energy which is lost at the interference is regained at the constructive interference. One wave is travelling horizontally, and the other is travelling downwards at an angle θ to the first wave, assuming that the two waves are in phase at the point B, then the relative phase changes along the x-axis. Constructive interference occurs when the waves are in phase, and destructive interference when they are half a cycle out of phase. Thus, a fringe pattern is produced, where the separation of the maxima is d f = λ sin θ
6.
Quantum decoherence
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Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons behave like waves and are described by a wavefunction and these waves can interfere, leading to the peculiar behaviour of quantum particles. As long as there exists a definite relation between different states, the system is said to be coherent. This coherence is a property of quantum mechanics, and is necessary for the function of quantum computers. However, when a system is not perfectly isolated, but in contact with its surroundings, the coherence decays with time. As a result of this process, the behaviour is lost. Decoherence was first introduced in 1970 by the German physicist H. Dieter Zeh and has been a subject of research since the 1980s. Decoherence can be viewed as the loss of information from a system into the environment, viewed in isolation, the systems dynamics are non-unitary. Thus the dynamics of the system alone are irreversible, as with any coupling, entanglements are generated between the system and environment. These have the effect of sharing quantum information with—or transferring it to—the surroundings, Decoherence has been used to understand the collapse of the wavefunction in quantum mechanics. Decoherence does not generate actual wave function collapse and it only provides an explanation for the observation of wave function collapse, as the quantum nature of the system leaks into the environment. That is, components of the wavefunction are decoupled from a coherent system, a total superposition of the global or universal wavefunction still exists, but its ultimate fate remains an interpretational issue. Specifically, decoherence does not attempt to explain the measurement problem, rather, decoherence provides an explanation for the transition of the system to a mixture of states that seem to correspond to those states observers perceive. Decoherence represents a challenge for the realization of quantum computers. Simply put, they require that coherent states be preserved and that decoherence is managed, to examine how decoherence operates, an intuitive model is presented. The model requires some familiarity with quantum theory basics, analogies are made between visualisable classical phase spaces and Hilbert spaces. A more rigorous derivation in Dirac notation shows how decoherence destroys interference effects, next, the density matrix approach is presented for perspective. An N-particle system can be represented in non-relativistic quantum mechanics by a wavefunction, ψ and this has analogies with the classical phase space
7.
Quantum entanglement
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Measurements of physical properties such as position, momentum, spin, and polarization, performed on entangled particles are found to be appropriately correlated. Later, however, the predictions of quantum mechanics were verified experimentally. Recent experiments have measured entangled particles within less than one hundredth of a percent of the time of light between them. According to the formalism of theory, the effect of measurement happens instantly. It is not possible, however, to use this effect to transmit information at faster-than-light speeds. Research is also focused on the utilization of entanglement effects in communication and computation, the counterintuitive predictions of quantum mechanics about strongly correlated systems were first discussed by Albert Einstein in 1935, in a joint paper with Boris Podolsky and Nathan Rosen. In this study, they formulated the EPR paradox, an experiment that attempted to show that quantum mechanical theory was incomplete. They wrote, We are thus forced to conclude that the description of physical reality given by wave functions is not complete. However, they did not coin the word entanglement, nor did they generalize the special properties of the state they considered and he shortly thereafter published a seminal paper defining and discussing the notion, and terming it entanglement. Like Einstein, Schrödinger was dissatisfied with the concept of entanglement, Einstein later famously derided entanglement as spukhafte Fernwirkung or spooky action at a distance. The EPR paper generated significant interest among physicists and inspired much discussion about the foundations of quantum mechanics, until recently each had left open at least one loophole by which it was possible to question the validity of the results. However, in 2015 the first loophole-free experiment was performed, which ruled out a class of local realism theories with certainty. The work of Bell raised the possibility of using these super-strong correlations as a resource for communication and it led to the discovery of quantum key distribution protocols, most famously BB84 by Charles H. Bennett and Gilles Brassard and E91 by Artur Ekert. Although BB84 does not use entanglement, Ekerts protocol uses the violation of a Bells inequality as a proof of security, in entanglement, one constituent cannot be fully described without considering the other. Quantum systems can become entangled through various types of interactions, for some ways in which entanglement may be achieved for experimental purposes, see the section below on methods. Entanglement is broken when the entangled particles decohere through interaction with the environment, for example, as an example of entanglement, a subatomic particle decays into an entangled pair of other particles. For instance, a particle could decay into a pair of spin-½ particles. The special property of entanglement can be observed if we separate the said two particles
8.
Energy level
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A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy. This contrasts with classical particles, which can have any energy and these discrete values are called energy levels. The energy spectrum of a system with discrete energy levels is said to be quantized. In chemistry and atomic physics, a shell, or a principal energy level. The closest shell to the nucleus is called the 1 shell, followed by the 2 shell, then the 3 shell, the shells correspond with the principal quantum numbers or are labeled alphabetically with letters used in the X-ray notation. Each shell can contain only a number of electrons, The first shell can hold up to two electrons, the second shell can hold up to eight electrons, the third shell can hold up to 18. The general formula is that the nth shell can in principle hold up to 2 electrons, since electrons are electrically attracted to the nucleus, an atoms electrons will generally occupy outer shells only if the more inner shells have already been completely filled by other electrons. However, this is not a requirement, atoms may have two or even three incomplete outer shells. For an explanation of why electrons exist in these shells see electron configuration, if the potential energy is set to zero at infinite distance from the atomic nucleus or molecule, the usual convention, then bound electron states have negative potential energy. If an atom, ion, or molecule is at the lowest possible level, it. If it is at an energy level, it is said to be excited. If more than one quantum state is at the same energy. They are then called degenerate energy levels, quantized energy levels result from the relation between a particles energy and its wavelength. For a confined particle such as an electron in an atom, only stationary states with energies corresponding to integral numbers of wavelengths can exist, for other states the waves interfere destructively, resulting in zero probability density. Elementary examples that show mathematically how energy levels come about are the particle in a box, the first evidence of quantization in atoms was the observation of spectral lines in light from the sun in the early 1800s by Joseph von Fraunhofer and William Hyde Wollaston. The notion of levels was proposed in 1913 by Danish physicist Niels Bohr in the Bohr theory of the atom. The modern quantum mechanical theory giving an explanation of these levels in terms of the Schrödinger equation was advanced by Erwin Schrödinger and Werner Heisenberg in 1926. When the electron is bound to the atom in any closer value of n, assume there is one electron in a given atomic orbital in a hydrogen-like atom
9.
Quantum state
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In quantum physics, quantum state refers to the state of an isolated quantum system. A quantum state provides a probability distribution for the value of each observable, knowledge of the quantum state together with the rules for the systems evolution in time exhausts all that can be predicted about the systems behavior. A mixture of states is again a quantum state. Quantum states that cannot be written as a mixture of states are called pure quantum states. Mathematically, a quantum state can be represented by a ray in a Hilbert space over the complex numbers. The ray is a set of nonzero vectors differing by just a scalar factor, any of them can be chosen as a state vector to represent the ray. A unit vector is usually picked, but its phase factor can be chosen freely anyway, nevertheless, such factors are important when state vectors are added together to form a superposition. Hilbert space is a generalization of the ordinary Euclidean space and it all possible pure quantum states of the given system. If this Hilbert space, by choice of representation, is exhibited as a function space, a more complicated case is given by the spin part of a state vector | ψ ⟩ =12, which involves superposition of joint spin states for two particles with spin 1⁄2. A mixed quantum state corresponds to a mixture of pure states, however. Mixed states are described by so-called density matrices, a pure state can also be recast as a density matrix, in this way, pure states can be represented as a subset of the more general mixed states. For example, if the spin of an electron is measured in any direction, e. g. with a Stern–Gerlach experiment, the Hilbert space for the electrons spin is therefore two-dimensional. A mixed state, in case, is a 2 ×2 matrix that is Hermitian, positive-definite. These probability distributions arise for both mixed states and pure states, it is impossible in quantum mechanics to prepare a state in all properties of the system are fixed. This is exemplified by the uncertainty principle, and reflects a difference between classical and quantum physics. Even in quantum theory, however, for every observable there are states that have an exact. In the mathematical formulation of mechanics, pure quantum states correspond to vectors in a Hilbert space. The operator serves as a function which acts on the states of the system
10.
Quantum superposition
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Quantum superposition is a fundamental principle of quantum mechanics. Mathematically, it refers to a property of solutions to the Schrödinger equation, since the Schrödinger equation is linear, an example of a physically observable manifestation of superposition is interference peaks from an electron wave in a double-slit experiment. Another example is a logical qubit state, as used in quantum information processing. Here |0 ⟩ is the Dirac notation for the state that will always give the result 0 when converted to classical logic by a measurement. Likewise |1 ⟩ is the state that will convert to 1. The numbers that describe the amplitudes for different possibilities define the kinematics, the dynamics describes how these numbers change with time. This list is called the vector, and formally it is an element of a Hilbert space. The quantities that describe how they change in time are the transition probabilities K x → y, which gives the probability that, starting at x, the particle ends up at y time t later. When no time passes, nothing changes, for 0 elapsed time K x → y = δ x y, the K matrix is zero except from a state to itself. So in the case that the time is short, it is better to talk about the rate of change of the probability instead of the change in the probability. Quantum amplitudes give the rate at which amplitudes change in time, the reason it is multiplied by i is that the condition that U is unitary translates to the condition, = I H † − H =0 which says that H is Hermitian. The eigenvalues of the Hermitian matrix H are real quantities, which have an interpretation as energy levels. For a particle that has equal amplitude to move left and right, the Hermitian matrix H is zero except for nearest neighbors, where it has the value c. If the coefficient is constant, the condition that H is Hermitian demands that the amplitude to move to the left is the complex conjugate of the amplitude to move to the right. By redefining the phase of the wavefunction in time, ψ → ψ e i 2 c t, but this phase rotation introduces a linear term. I d ψ n d t = c ψ n +1 −2 c ψ n + c ψ n −1, the analogy between quantum mechanics and probability is very strong, so that there are many mathematical links between them. The analogous expression in quantum mechanics is the path integral, a generic transition matrix in probability has a stationary distribution, which is the eventual probability to be found at any point no matter what the starting point. If there is a probability for any two paths to reach the same point at the same time, this stationary distribution does not depend on the initial conditions
11.
Symmetry in quantum mechanics
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In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are methods for solving problems and predicting what can happen. While conservation laws do not always give the answer to the problem directly, they form the correct constraints, the notational conventions used in this article are as follows. Boldface indicates vectors, four vectors, matrices, and vectorial operators, wide hats are for operators, narrow hats are for unit vectors. The summation convention on the repeated indices is used, unless stated otherwise. Generally, the correspondence between continuous symmetries and conservation laws is given by Noethers theorem and this can be done for displacements, durations, and angles. Additionally, the invariance of certain quantities can be seen by making changes in lengths and angles. In what follows, transformations on only one-particle wavefunctions in the form, Ω ^ ψ = ψ are considered, unitarity is generally required for operators representing transformations of space, time, and spin, since the norm of a state must be invariant under these transformations. The inverse is the Hermitian conjugate Ω ^ −1 = Ω ^ †, the results can be extended to many-particle wavefunctions. Quantum operators representing observables are also required to be Hermitian so that their eigenvalues are real numbers, i. e. the operator equals its Hermitian conjugate, following are the key points of group theory relevant to quantum theory, examples are given throughout the article. For an alternative approach using matrix groups, see the books of Hall Let G be a Lie group, ξN. the dimension of the group, N, is the number of parameters it has. The generators satisfy the commutator, = i f a b c X c where fabc are the constants of the group. This makes, together with the vector space property, the set of all generators of a group a Lie algebra, due to the antisymmetry of the bracket, the structure constants of the group are antisymmetric in the first two indices. The representations of the group are denoted using a capital D and defined by, representations are linear operators that take in group elements and preserve the composition rule, D D = D. A representation which cannot be decomposed into a sum of other representations, is called irreducible. It is conventional to label irreducible representations by a number n in brackets, as in D, or if there is more than one number. Representations also exist for the generators and the notation of a capital D is used in this context. An example of abuse is to be found in the defining equation above
12.
Quantum tunnelling
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Quantum tunnelling or tunneling refers to the quantum mechanical phenomenon where a particle tunnels through a barrier that it classically could not surmount. This plays an role in several physical phenomena, such as the nuclear fusion that occurs in main sequence stars like the Sun. It has important applications to modern devices such as the diode, quantum computing. The effect was predicted in the early 20th century and its acceptance as a physical phenomenon came mid-century. Tunnelling is often explained using the Heisenberg uncertainty principle and the duality of matter. Pure quantum mechanical concepts are central to the phenomenon, so quantum tunnelling is one of the implications of quantum mechanics. Quantum tunnelling was developed from the study of radioactivity, which was discovered in 1896 by Henri Becquerel, radioactivity was examined further by Marie Curie and Pierre Curie, for which they earned the Nobel Prize in Physics in 1903. Ernest Rutherford and Egon Schweidler studied its nature, which was later verified empirically by Friedrich Kohlrausch, the idea of the half-life and the impossibility of predicting decay was created from their work. J. J. Thomson commented the finding warranted further investigation, in 1926, Rother, using a still newer platform galvanometer of sensitivity 26 pA, measured the field emission currents in a hard vacuum between closely spaced electrodes. Friedrich Hund was the first to notice of tunnelling in 1927 when he was calculating the ground state of the double-well potential. Its first application was an explanation for alpha decay, which was done in 1928 by George Gamow and independently by Ronald Gurney. After attending a seminar by Gamow, Max Born recognised the generality of tunnelling and he realised that it was not restricted to nuclear physics, but was a general result of quantum mechanics that applies to many different systems. Shortly thereafter, both considered the case of particles tunnelling into the nucleus. The study of semiconductors and the development of transistors and diodes led to the acceptance of electron tunnelling in solids by 1957. The work of Leo Esaki, Ivar Giaever and Brian Josephson predicted the tunnelling of superconducting Cooper pairs, in 2016, the quantum tunneling of water was discovered. Quantum tunnelling falls under the domain of quantum mechanics, the study of what happens at the quantum scale and this process cannot be directly perceived, but much of its understanding is shaped by the microscopic world, which classical mechanics cannot adequately explain. Classical mechanics predicts that particles that do not have enough energy to surmount a barrier will not be able to reach the other side. Thus, a ball without sufficient energy to surmount the hill would roll back down, or, lacking the energy to penetrate a wall, it would bounce back or in the extreme case, bury itself inside the wall
13.
Uncertainty principle
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The formal inequality relating the standard deviation of position σx and the standard deviation of momentum σp was derived by Earle Hesse Kennard later that year and by Hermann Weyl in 1928. Heisenberg offered such an effect at the quantum level as a physical explanation of quantum uncertainty. Thus, the uncertainty principle actually states a fundamental property of quantum systems, since the uncertainty principle is such a basic result in quantum mechanics, typical experiments in quantum mechanics routinely observe aspects of it. Certain experiments, however, may deliberately test a particular form of the uncertainty principle as part of their research program. These include, for example, tests of number–phase uncertainty relations in superconducting or quantum optics systems, applications dependent on the uncertainty principle for their operation include extremely low-noise technology such as that required in gravitational wave interferometers. The uncertainty principle is not readily apparent on the scales of everyday experience. So it is helpful to demonstrate how it applies to more easily understood physical situations, two alternative frameworks for quantum physics offer different explanations for the uncertainty principle. The wave mechanics picture of the uncertainty principle is more visually intuitive, a nonzero function and its Fourier transform cannot both be sharply localized. In matrix mechanics, the formulation of quantum mechanics, any pair of non-commuting self-adjoint operators representing observables are subject to similar uncertainty limits. An eigenstate of an observable represents the state of the wavefunction for a certain measurement value, for example, if a measurement of an observable A is performed, then the system is in a particular eigenstate Ψ of that observable. According to the de Broglie hypothesis, every object in the universe is a wave, the position of the particle is described by a wave function Ψ. The time-independent wave function of a plane wave of wavenumber k0 or momentum p0 is ψ ∝ e i k 0 x = e i p 0 x / ℏ. In the case of the plane wave, | ψ |2 is a uniform distribution. In other words, the position is extremely uncertain in the sense that it could be essentially anywhere along the wave packet. The figures to the right show how with the addition of many plane waves, in mathematical terms, we say that ϕ is the Fourier transform of ψ and that x and p are conjugate variables. Adding together all of these plane waves comes at a cost, namely the momentum has become less precise, One way to quantify the precision of the position and momentum is the standard deviation σ. Since | ψ |2 is a probability density function for position, the precision of the position is improved, i. e. reduced σx, by using many plane waves, thereby weakening the precision of the momentum, i. e. increased σp. Another way of stating this is that σx and σp have a relationship or are at least bounded from below
14.
Wave function
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A wave function in quantum physics is a description of the quantum state of a system. The wave function is a probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a function are the Greek letters ψ or Ψ. The wave function is a function of the degrees of freedom corresponding to some set of commuting observables. Once such a representation is chosen, the function can be derived from the quantum state. For a given system, the choice of which commuting degrees of freedom to use is not unique, some particles, like electrons and photons, have nonzero spin, and the wave function for such particles includes spin as an intrinsic, discrete degree of freedom. Other discrete variables can also be included, such as isospin and these values are often displayed in a column matrix. According to the principle of quantum mechanics, wave functions can be added together and multiplied by complex numbers to form new wave functions. The Schrödinger equation determines how wave functions evolve over time, a wave function behaves qualitatively like other waves, such as water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation. This explains the name wave function, and gives rise to wave–particle duality, however, the wave function in quantum mechanics describes a kind of physical phenomenon, still open to different interpretations, which fundamentally differs from that of classic mechanical waves. The integral of this quantity, over all the degrees of freedom. This general requirement a wave function must satisfy is called the normalization condition, since the wave function is complex valued, only its relative phase and relative magnitude can be measured. In 1905 Einstein postulated the proportionality between the frequency of a photon and its energy, E = hf, and in 1916 the corresponding relation between photon momentum and wavelength, λ = h/p, the equations represent wave–particle duality for both massless and massive particles. In the 1920s and 1930s, quantum mechanics was developed using calculus and those who used the techniques of calculus included Louis de Broglie, Erwin Schrödinger, and others, developing wave mechanics. Those who applied the methods of linear algebra included Werner Heisenberg, Max Born, Schrödinger subsequently showed that the two approaches were equivalent. However, no one was clear on how to interpret it, at first, Schrödinger and others thought that wave functions represent particles that are spread out with most of the particle being where the wave function is large. This was shown to be incompatible with the scattering of a wave packet representing a particle off a target. While a scattered particle may scatter in any direction, it not break up
15.
Afshar experiment
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The Afshar experiment is an optics experiment, devised and carried out by Shahriar Afshar at Harvard University in 2004, which is a variation of the double slit experiment in quantum mechanics. Afshars experiment uses a variant of Thomas Youngs classic double-slit experiment to create patterns to investigate complementarity. Such interferometer experiments typically have two arms or paths a photon may take, one of Afshars assertions is that, in his experiment, it is possible to check for interference fringes of a photon stream while at the same time observing each photons path. The results were presented at a Harvard seminar in March 2004, the experiment was featured as the cover story in the July 24,2004 edition of New Scientist. Afshar presented his work also at the American Physical Society meeting in Los Angeles and his peer-reviewed paper was published in Foundations of Physics in January 2007. Afshar claims that his experiment invalidates the complementarity principle and has far-reaching implications for the understanding of quantum mechanics, according to Cramer, Afshars results support Cramers own transactional interpretation of quantum mechanics and challenge the many-worlds interpretation of quantum mechanics. This claim has not been published in a reviewed journal. The experiment uses a similar to that for the double-slit experiment. In Afshars variant, light generated by a laser passes through two closely spaced circular pinholes, after the dual pinholes, a lens refocuses the light so that the image of each pinhole falls on separate photon-detectors. When the light acts as a wave, because of quantum interference one can observe that there are regions that the photons avoid, called dark fringes. A grid of wires is placed just before the lens so that the wires lie in the dark fringes of an interference pattern which is produced by the dual pinhole setup. If one of the pinholes is blocked, the pattern will no longer be formed. Consequently, the quality is reduced. When one pinhole is closed, the grid of wires causes appreciable diffraction in the light, the effect is not dependent on the light intensity. Afshar argues that this contradicts the principle of complementarity, since it shows both complementary wave and particle characteristics in the same experiment for the same photons. Afshar has responded to critics in his academic talks, his blog. She proposes that Afshars experiment is equivalent to preparing an electron in a spin-up state and this does not imply that one has found out the up-down spin state and the sideways spin state of any electron simultaneously. In addition she underscores her conclusion with an analysis of the Afshar setup within the framework of the interpretation of quantum mechanics
16.
Bell test experiments
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Under local realism, correlations between outcomes of different measurements performed on separated physical systems have to satisfy certain constraints, called Bell inequalities. John Bell derived the first inequality of this kind in his paper On the Einstein-Podolsky-Rosen Paradox, Bells Theorem states that the predictions of quantum mechanics, concerning correlations, being inconsistent with Bells inequality, cannot be reproduced by any local hidden variable theory. However, this doesnt disprove hidden variable theories that are such as Bohmian mechanics. A Bell test experiment is one designed to test whether or not the real world satisfies local realism, in practice most actual experiments have used light, assumed to be emitted in the form of particle-like photons, rather than the atoms that Bell originally had in mind. The property of interest is, in the best known experiments, such experiments fall into two classes, depending on whether the analysers used have one or two output channels. The diagram shows an optical experiment of the two-channel kind for which Alain Aspect set a precedent in 1982. Coincidences are recorded, the results being categorised as ++, +−, −+ or −−, four separate subexperiments are conducted, corresponding to the four terms E in the test statistic S. For each selected value of a and b, the numbers of coincidences in each category are recorded, the experimental estimate for E is then calculated as, E = /. Once all four E’s have been estimated, an estimate of the test statistic S = E − E + E + E can be found. If S is numerically greater than 2 it has infringed the CHSH inequality, the experiment is declared to have supported the QM prediction and ruled out all local hidden variable theories. A strong assumption has had to be made, however, to use of expression. It has been assumed that the sample of detected pairs is representative of the emitted by the source. That this assumption may not be true comprises the fair sampling loophole, the derivation of the inequality is given in the CHSH Bell test page. Prior to 1982 all actual Bell tests used single-channel polarisers and variations on an inequality designed for this setup, the latter is described in Clauser, Horne, Shimony and Holts much-cited 1969 article as being the one suitable for practical use. Counts are taken as before and used to estimate the test statistic, S = / N, where the symbol ∞ indicates absence of a polariser. If S exceeds 0 then the experiment is declared to have infringed Bells inequality, in order to derive, CHSH in their 1969 paper had to make an extra assumption, the so-called fair sampling assumption. This means that the probability of detection of a given photon, if this assumption were violated, then in principle a local hidden variable model could violate the CHSH inequality. In a later 1974 article, Clauser and Horne replaced this assumption by a weaker, no enhancement assumption, deriving a modified inequality, see the page on Clauser
17.
Double-slit experiment
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A simpler form of the double-slit experiment was performed originally by Thomas Young in 1801. He believed it demonstrated that the theory of light was correct. The experiment belongs to a class of double path experiments. Changes in the lengths of both waves result in a phase shift, creating an interference pattern. Another version is the Mach–Zehnder interferometer, which splits the beam with a mirror, furthermore, versions of the experiment that include detectors at the slits find that each detected photon passes through one slit, and not through both slits. However, such experiments demonstrate that particles do not form the pattern if one detects which slit they pass through. These results demonstrate the principle of wave–particle duality, other atomic-scale entities, such as electrons, are found to exhibit the same behavior when fired towards a double slit. Additionally, the detection of individual impacts is observed to be inherently probabilistic. The experiment can be done with much larger than electrons and photons. The largest entities for which the experiment has been performed were molecules that each comprised 810 atoms. However, when this experiment is actually performed, the pattern on the screen is a diffraction pattern in which the light is spread out. The smaller the slit, the greater the angle of spread, the top portion of the image shows the central portion of the pattern formed when a red laser illuminates a slit and, if one looks carefully, two faint side bands. More bands can be seen with a highly refined apparatus. Diffraction explains the pattern as being the result of the interference of waves from the slit. If one illuminates two parallel slits, the light from the two slits again interferes, here the interference is a more-pronounced pattern with a series of light and dark bands. The width of the bands is a property of the frequency of the illuminating light, however, the later discovery of the photoelectric effect demonstrated that under different circumstances, light can behave as if it is composed of discrete particles. These seemingly contradictory discoveries made it necessary to go beyond classical physics, the double-slit experiment has become a classic thought experiment, for its clarity in expressing the central puzzles of quantum mechanics. In reality, it contains the only mystery, Feynman was fond of saying that all of quantum mechanics can be gleaned from carefully thinking through the implications of this single experiment
18.
Popper's experiment
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Poppers experiment is an experiment proposed by the philosopher Karl Popper to put to the test different interpretations of quantum mechanics. In fact, as early as 1934, Popper started criticising the increasingly more accepted Copenhagen interpretation, nonetheless, Popper gave other and important contributions to the foundations of quantum mechanics in different periods of his long and prolific career. In particular, in the 1980s, he established collaborations and new acquaintances with some illustrious physicists working in the field of foundations of QM, in 1980, Popper proposed his more important, yet overlooked, contribution to QM, a new simplified version of the EPR experiment. The experiment was published only two years later, in the third volume of the Poscript to the Logic of Scientific Discovery. The most widely known interpretation of quantum mechanics is the Copenhagen interpretation put forward by Niels Bohr and it maintains that observations lead to a wavefunction collapse, thereby suggesting the counter-intuitive result that two well separated, non-interacting systems require action-at-a-distance. Popper argued that such non-locality conflicts with common sense, and would lead to a subjectivist interpretation of phenomena, contrarily to the first proposal of 1934, Poppers experiment of 1980 exploits couples of entangled particles, in order to put to the test Heisenbergs uncertainty principle. Poppers proposed experiment consists of a low-intensity source of particles that can generate pairs of particles traveling to the left and to the right along the x-axis. The beams low intensity is so that the probability is high that two particles recorded at the time on the left and on the right are those which have actually interacted before emission. There are two slits, one each in the paths of the two particles, behind the slits are semicircular arrays of counters which can detect the particles after they pass through the slits. These counters are coincident counters that they only detect particles that have passed at the time through A and B. This larger spread in the momentum will show up as particles being detected even at positions that lie outside the regions where particles would normally reach based on their initial momentum spread. Popper suggests that we count the particles in coincidence, i. e. we count only those particles behind slit B, particles which are not able to pass through slit A are ignored. The Heisenberg scatter for both the beams of particles going to the right and to the left, is tested by making the two slits A and B wider or narrower. If the slits are narrower, then counters should come into play which are higher up and lower down, the coming into play of these counters is indicative of the wider scattering angles which go with a narrower slit, according to the Heisenberg relations. Now the slit at A is made very small and the slit at B very wide, Popper wrote that, according to the EPR argument, we have measured position y for both particles with the precision Δ y, and not just for particle passing through slit A. This is because from the initial entangled EPR state we can calculate the position of the particle 2, once the position of particle 1 is known and we can do this, argues Popper, even though slit B is wide open. Therefore, Popper states that fairly precise knowledge about the y position of particle 2 is made, now the scatter can, in principle, be tested with the help of the counters. Popper was inclined to believe that the test would decide against the Copenhagen interpretation, if the test decided in favor of the Copenhagen interpretation, Popper argued, it could be interpreted as indicative of action at a distance
19.
Quantum eraser experiment
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Next, the experimenter marks through which slit each photon went and demonstrates that thereafter the interference pattern is destroyed. This stage indicates that it is the existence of the information that causes the destruction of the interference pattern. Third, the information is erased, whereupon the interference pattern is recovered. A key result is that it does not matter whether the procedure is done before or after the photons arrive at the detection screen. Quantum erasure technology can be used to increase the resolution of advanced microscopes, the quantum eraser experiment described in this article is a variation of Thomas Youngs classic double-slit experiment. It establishes that when action is taken to determine which slit a photon has passed through, when a stream of photons is marked in this way, then the interference fringes characteristic of the Young experiment will not be seen. The experiment described in this article is capable of creating situations in which a photon that has been marked to reveal through which slit it has passed can later be unmarked and this experiment involves an apparatus with two main sections. After two entangled photons are created, each is directed into its own section of the apparatus, anything done to learn the path of the entangled partner of the photon being examined in the double-slit part of the apparatus will influence the second photon, and vice versa. In doing so, the experimenter restores interference without altering the part of the experimental apparatus. In delayed-choice experiments quantum effects can mimic an influence of future actions on past events, however, the temporal order of measurement actions is not relevant. First, a photon is shot through a nonlinear optical device. This crystal converts the photon into two entangled photons of lower frequency, a process known as spontaneous parametric down-conversion. These entangled photons follow separate paths, one photon goes directly to a detector, while the second photon passes through the double-slit mask to a second detector. Both detectors are connected to a circuit, ensuring that only entangled photon pairs are counted. A stepper motor moves the second detector to scan across the target area and this configuration yields the familiar interference pattern. This polarization is measured at the detector, thus marking the photons, finally, a linear polarizer is introduced in the path of the first photon of the entangled pair, giving this photon a diagonal polarization. Entanglement ensures a complementary diagonal polarization in its partner, which passes through the double-slit mask and this alters the effect of the circular polarizers, each will produce a mix of clockwise and counter-clockwise polarized light. Thus the second detector can no longer determine which path was taken, a double slit with rotating polarizers can also be accounted for by considering the light to be a classical wave
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Delayed choice quantum eraser
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The experiment was designed to investigate peculiar consequences of the well-known double-slit experiment in quantum mechanics, as well as the consequences of quantum entanglement. The delayed choice quantum eraser experiment investigates a paradox, If a photon manifests itself as though it had come by a single path to the detector, then common sense says it must have entered the double-slit device as a particle. If a photon manifests itself as though it had come by two paths, then it must have entered the double-slit device as a wave. If the experimental apparatus is changed while the photon is in mid‑flight and this is the standard view, and recent experiments have supported it. In the basic double slit experiment, a beam of light is directed perpendicularly towards a wall pierced by two parallel slit apertures. If a detection screen is put on the side of the double slit wall, a pattern of light and dark fringes will be observed. Other atomic-scale entities such as electrons are found to exhibit the same behavior when fired toward a double slit, by decreasing the brightness of the source sufficiently, individual particles that form the interference pattern are detectable. This is an idea that contradicts our everyday experience of discrete objects and this which-way experiment illustrates the complementarity principle that photons can behave as either particles or waves, but not both at the same time. However, technically feasible realizations of this experiment were not proposed until the 1970s, which-path information and the visibility of interference fringes are hence complementary quantities. However, in 1982, Scully and Drühl found a loophole around this interpretation and they proposed a quantum eraser to obtain which-path information without scattering the particles or otherwise introducing uncontrolled phase factors to them. Lest there be any misunderstanding, the pattern does disappear when the photons are so marked. However, the interference pattern if the which-path information is further manipulated after the marked photons have passed through the double slits to obscure the which-path markings. Since 1982, multiple experiments have demonstrated the validity of the quantum eraser. A simple version of the eraser can be described as follows. In the two diagrams in Fig.1, photons are emitted one at a time from a laser symbolized by a yellow star and they pass through a 50% beam splitter that reflects or transmits 1/2 of the photons. The reflected or transmitted photons travel along two possible paths depicted by the red or blue lines, in the bottom diagram, a second beam splitter is introduced at the top right. It can direct either beam toward either exit port, thus, photons emerging from each exit port may have come by way of either path. By introducing the second beam splitter, the information has been erased
21.
Wheeler's delayed choice experiment
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Wheelers delayed choice experiment is actually several thought experiments in quantum physics, proposed by John Archibald Wheeler, with the most prominent among them appearing in 1978 and 1984. Some interpreters of these experiments contend that a photon either is a wave or is a particle, Wheelers intent was to investigate the time-related conditions under which a photon makes this transition between alleged states of being. His work has been productive of many revealing experiments, however, he himself seems to be very clear on this point. Either it was a wave or a particle, either it went both ways around the galaxy or only one way, actually, quantum phenomena are neither waves nor particles but are intrinsically undefined until the moment they are measured. In a sense, the British philosopher Bishop Berkeley was right when he asserted two centuries ago to be is to be perceived and this line of experimentation proved very difficult to carry out when it was first conceived. Nevertheless, it has proven very valuable over the years since it has led researchers to provide increasingly sophisticated demonstrations of the duality of single quanta. As one experimenter explains, Wave and particle behavior can coexist simultaneously, Wheelers delayed choice experiment refers to a series of thought experiments in quantum physics, the first being proposed by him in 1978. Another prominent version was proposed in 1983, all of these experiments try to get at the same fundamental issues in quantum physics. According to the complementarity principle, a photon can manifest properties of a particle or of a wave, what characteristic is manifested depends on whether experimenters use a device intended to observe particles or to observe waves. When this statement is applied very strictly, one could argue that by determining the type one could force the photon to become manifest only as a particle or only as a wave. Detection of a photon is a process because a photon can never be seen in flight. A photon always appears at some highly localized point in space, suppose that a traditional double-slit experiment is prepared so that either of the slits can be blocked. If both slits are open and a series of photons are emitted by the then a interference pattern will quickly emerge on the detection screen. The interference pattern can only be explained as a consequence of wave phenomena, if only one slit is available then there will be no interference pattern, so experimenters may conclude that each photon decides to travel as a particle as soon as it is emitted. One way to investigate the question of when a photon decides whether to act as a wave or a particle in an experiment is to use the interferometer method. If the apparatus is changed so that a beam splitter is placed in the upper-right corner. Experimenters must explain these phenomena as consequences of the nature of light. They may affirm that each photon must have traveled by both paths as a wave else that photon could not have interfered with itself
22.
Mathematical formulation of quantum mechanics
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The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. Many of these structures are drawn from functional analysis, an area within pure mathematics that was influenced in part by the needs of quantum mechanics. These formulations of quantum mechanics continue to be used today, at the heart of the description are ideas of quantum state and quantum observable which are radically different from those used in previous models of physical reality. While the mathematics permits calculation of many quantities that can be measured experimentally, probability theory was used in statistical mechanics. Geometric intuition played a role in the first two and, accordingly, theories of relativity were formulated entirely in terms of geometric concepts. The most sophisticated example of this is the Sommerfeld–Wilson–Ishiwara quantization rule, planck postulated a direct proportionality between the frequency of radiation and the quantum of energy at that frequency. The proportionality constant, h, is now called Plancks constant in his honor, in 1905, Einstein explained certain features of the photoelectric effect by assuming that Plancks energy quanta were actual particles, which were later dubbed photons. All of these developments were phenomenological and challenged the theoretical physics of the time, Bohr and Sommerfeld went on to modify classical mechanics in an attempt to deduce the Bohr model from first principles. The most sophisticated version of formalism was the so-called Sommerfeld–Wilson–Ishiwara quantization. Although the Bohr model of the hydrogen atom could be explained in this way, the mathematical status of quantum theory remained uncertain for some time. In 1923 de Broglie proposed that wave–particle duality applied not only to photons but to electrons, the physical interpretation of the theory was also clarified in these years after Werner Heisenberg discovered the uncertainty relations and Niels Bohr introduced the idea of complementarity. Werner Heisenbergs matrix mechanics was the first successful attempt at replicating the observed quantization of atomic spectra, later in the same year, Schrödinger created his wave mechanics. Schrödingers formalism was considered easier to understand, visualize and calculate as it led to differential equations, within a year, it was shown that the two theories were equivalent. It was Max Born who introduced the interpretation of the square of the wave function as the probability distribution of the position of a pointlike object. Borns idea was taken over by Niels Bohr in Copenhagen who then became the father of the Copenhagen interpretation of quantum mechanics. Schrödingers wave function can be seen to be related to the classical Hamilton–Jacobi equation. The correspondence to classical mechanics was even more explicit, although somewhat more formal, in fact, in these early years, linear algebra was not generally popular with physicists in its present form. He is the third, and possibly most important, pillar of that field and his work was particularly fruitful in all kinds of generalizations of the field
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Phase space formulation
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The phase space formulation of quantum mechanics places the position and momentum variables on equal footing, in phase space. In contrast, the Schrödinger picture uses the position or momentum representations, the two key features of the phase space formulation are that the quantum state is described by a quasiprobability distribution and operator multiplication is replaced by a star product. The theory was developed by Hilbrand Groenewold in 1946 in his PhD thesis. This formulation is statistical in nature and offers connections between quantum mechanics and classical statistical mechanics, enabling a natural comparison between the two. The conceptual ideas underlying the development of mechanics in phase space have branched into mathematical offshoots such as algebraic deformation theory. The phase space distribution f of a state is a quasiprobability distribution. There are several different ways to represent the distribution, all interrelated, the most noteworthy is the Wigner representation, W, discovered first. Other representations include the Glauber-Sudarshan P, Husimi Q, Kirkwood-Rihaczek, Mehta, Rivier and these alternatives are most useful when the Hamiltonian takes a particular form, such as normal order for the Glauber–Sudarshan P-representation. Since the Wigner representation is the most common, this article will usually stick to it, the phase space distribution possesses properties akin to the probability density in a 2n-dimensional phase space. For example, it is real-valued, unlike the generally complex-valued wave function. We can understand the probability of lying within an interval, for example, by integrating the Wigner function over all momenta and over the position interval. If Â is an operator representing an observable, it may be mapped to phase space as A through the Wigner transform, conversely, this operator may be recovered via the Weyl transform. The expectation value of the observable with respect to the phase space distribution is ⟨ A ^ ⟩ = ∫ A W d p d x. Moreover, it can, in general, take negative values even for states, with the unique exception of coherent states. Regions of such negative value are provable to be small, they extend to compact regions larger than a few ħ. They are shielded by the uncertainty principle, which does not allow precise localization within phase-space regions smaller than ħ, the fundamental noncommutative binary operator in the phase space formulation that replaces the standard operator multiplication is the star product, represented by the symbol ★. Each representation of the distribution has a different characteristic star product. For concreteness, we restrict this discussion to the star product relevant to the Wigner-Weyl representation, for notational convenience, we introduce the notion of left and right derivatives
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Path integral formulation
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The path integral formulation of quantum mechanics is a description of quantum theory that generalizes the action principle of classical mechanics. Unlike previous methods, the path integral allows a physicist to easily change coordinates between very different canonical descriptions of the quantum system. Another advantage is that it is in easier to guess the correct form of the Lagrangian of a theory. Possible downsides of the approach include that unitarity of the S-matrix is obscure in the formulation, the path-integral approach has been proved to be equivalent to the other formalisms of quantum mechanics and quantum field theory. Thus, by deriving either approach from the other, problems associated with one or the other approach go away. The Schrödinger equation is an equation with an imaginary diffusion constant. The basic idea of the integral formulation can be traced back to Norbert Wiener. This idea was extended to the use of the Lagrangian in quantum mechanics by P. A. M. Dirac in his 1933 article, the complete method was developed in 1948 by Richard Feynman. Some preliminaries were worked out earlier in his work under the supervision of John Archibald Wheeler. The original motivation stemmed from the desire to obtain a quantum-mechanical formulation for the Wheeler–Feynman absorber theory using a Lagrangian as a starting point, in quantum mechanics, as in classical mechanics, the Hamiltonian is the generator of time translations. This means that the state at a later time differs from the state at the current time by the result of acting with the Hamiltonian operator. For states with an energy, this is a statement of the de Broglie relation between frequency and energy, and the general relation is consistent with that plus the superposition principle. The Hamiltonian in classical mechanics is derived from a Lagrangian, which is a fundamental quantity relative to special relativity. The Hamiltonian indicates how to march forward in time, but the time is different in different reference frames, so the Hamiltonian is different in different frames, and this type of symmetry is not apparent in the original formulation of quantum mechanics. The Hamiltonian is a function of the position and momentum at one time, the Lagrangian is a function of the position now and the position a little later. The relation between the two is by a Legendre transform, and the condition that determines the classical equations of motion is that the action has an extremum, in quantum mechanics, the Legendre transform is hard to interpret, because the motion is not over a definite trajectory. In classical mechanics, with discretization in time, the Legendre transform becomes ϵ H = p − ϵ L and p = ∂ L ∂ q ˙, where the partial derivative with respect to q ˙ holds q fixed. The inverse Legendre transform is ϵ L = ϵ p q ˙ − ϵ H, where q ˙ = ∂ H ∂ p, and the partial derivative now is with respect to p at fixed q
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Dirac equation
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In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles such as electrons and it was validated by accounting for the fine details of the hydrogen spectrum in a completely rigorous way. The equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved, moreover, in the limit of zero mass, the Dirac equation reduces to the Weyl equation. This accomplishment has been described as fully on a par with the works of Newton, Maxwell, in the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin-1/2 particles. The Dirac equation in the originally proposed by Dirac is. The p1, p2, p3 are the components of the momentum, also, c is the speed of light, and ħ is the Planck constant divided by 2π. These fundamental physical constants reflect special relativity and quantum mechanics, respectively, Diracs purpose in casting this equation was to explain the behavior of the relativistically moving electron, and so to allow the atom to be treated in a manner consistent with relativity. His rather modest hope was that the corrections introduced this way might have a bearing on the problem of atomic spectra, the new elements in this equation are the 4 ×4 matrices αk and β, and the four-component wave function ψ. There are four components in ψ because the evaluation of it at any point in configuration space is a bispinor. It is interpreted as a superposition of an electron, a spin-down electron, a spin-up positron. These matrices and the form of the function have a deep mathematical significance. The algebraic structure represented by the matrices had been created some 50 years earlier by the English mathematician W. K. Clifford. In turn, Cliffords ideas had emerged from the work of the German mathematician Hermann Grassmann in his Lineale Ausdehnungslehre. The latter had been regarded as well-nigh incomprehensible by most of his contemporaries, the appearance of something so seemingly abstract, at such a late date, and in such a direct physical manner, is one of the most remarkable chapters in the history of physics. The Dirac equation is similar to the Schrödinger equation for a massive free particle. The left side represents the square of the momentum operator divided by twice the mass, space and time derivatives both enter to second order. This has a consequence for the interpretation of the equation. Because the equation is second order in the derivative, one must specify initial values both of the wave function itself and of its first-time derivative in order to solve definite problems
26.
Rydberg formula
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The Rydberg formula is used in atomic physics to describe the wavelengths of spectral lines of many chemical elements. It was formulated by the Swedish physicist Johannes Rydberg, and presented on 5 November 1888, in 1880, Rydberg worked on a formula describing the relation between the wavelengths in spectral lines of alkali metals. He noticed that lines came in series and he found that he could simplify his calculations by using the wavenumber as his unit of measurement and he plotted the wavenumbers of successive lines in each series against consecutive integers which represented the order of the lines in that particular series. Finding that the curves were similarly shaped, he sought a single function which could generate all of them. This did not work very well, Rydberg therefore rewrote Balmers formula in terms of wavenumbers, as n = n 0 −4 n 0 m 2. This suggested that the Balmer formula for hydrogen might be a case with m ′ =0 and C0 =4 n 0, where n 0 =1 h. The term Co was found to be a universal constant common to all elements and this constant is now known as the Rydberg constant, and m is known as the quantum defect. As stressed by Niels Bohr, expressing results in terms of wavenumber, the fundamental role of wavenumbers was also emphasized by the Rydberg-Ritz combination principle of 1908. The fundamental reason for this lies in quantum mechanics, lights wavenumber is proportional to frequency 1 λ = f c, and therefore also proportional to lights quantum energy E. Thus,1 λ = E h c. Rydbergs 1888 classical expression for the form of the series was not accompanied by a physical explanation. In Bohrs conception of the atom, the integer Rydberg n numbers represent electron orbitals at different integral distances from the atom. A frequency emitted in a transition from n1 to n2 therefore represents the energy emitted or absorbed when an electron makes a jump from orbital 1 to orbital 2. Later models found that the values for n1 and n2 corresponded to the quantum numbers of the two orbitals. The number of protons in the nucleus of this element, n 1 and n 2 are integers such that n 1 < n 2, corresponding to the principal quantum numbers of the orbitals occupied before. Examples would include He+, Li2+, Be3+ etc. where no other electrons exist in the atom and this is analogous to the Lyman-alpha line transition for hydrogen, and has the same frequency factor. Its frequency is thus the Lyman-alpha hydrogen frequency, increased by a factor of 2. This formula of f = c/λ = ⋅2 is historically known as Moseleys law, see the biography of Henry Moseley for the historical importance of this law, which was derived empirically at about the same time it was explained by the Bohr model of the atom. Rydberg–Ritz combination principle Balmer series Hydrogen line Sutton, Mike, getting the numbers right, The lonely struggle of the 19th century physicist/chemist Johannes Rydberg
27.
Interpretations of quantum mechanics
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An interpretation of quantum mechanics is a set of statements which attempt to explain how quantum mechanics informs our understanding of nature. Although quantum mechanics has held up to rigorous and thorough experimental testing and this question is of special interest to philosophers of physics, as physicists continue to show a strong interest in the subject. The definition of quantum theorists terms, such as functions and matrix mechanics. Although the Copenhagen interpretation was originally most popular, quantum decoherence has gained popularity, thus the many-worlds interpretation has been gaining acceptance. The authors reference a similarly informal poll carried out by Max Tegmark at the Fundamental Problems in Quantum Theory conference in August 1997, in Tegmarks poll, the Everett interpretation received 17% of the vote, which is similar to the number of votes in our poll. A general law is a regularity of outcomes, whereas a causal mechanism may regulate the outcomes, a phenomenon can receive interpretation either ontic or epistemic. For instance, indeterminism may be attributed to limitations of human observation and perception, in a broad sense, scientific theory can be viewed as offering scientific realism—approximately true description or explanation of the natural world—or might be perceived with antirealism. A realist stance seeks the epistemic and the ontic, whereas an antirealist stance seeks epistemic, in the 20th centurys first half, antirealism was mainly logical positivism, which sought to exclude unobservable aspects of reality from scientific theory. The instrumentalist view is carried by the quote of David Mermin, Shut up and calculate. Other approaches to conceptual problems introduce new mathematical formalism. In classical field theory, a property at a given location in the field is readily derived. In Heisenbergs formalism, on the hand, to derive physical information about a location in the field, one must apply a quantum operation to a quantum state. Schrödingers formalism describes a waveform governing probability of outcomes across a field, yet how do we find in a specific location a particle whose wavefunction of mere probability distribution of existence spans a vast region of space. The act of measurement can interact with the state in peculiar ways. Yet quantum decoherence grants that all the possibilities can be real, Quantum entanglement, as illustrated in the EPR paradox, seemingly violates principles of local causality. Complementarity holds that no set of physical concepts can simultaneously refer to all properties of a quantum system. For instance, wave description A and particulate description B can each describe quantum system S, as now well known, the origin of complementarity lies in the non-commutativity of operators that describe quantum objects. Since the intricacy of a system is exponential, it is difficult to derive classical approximations
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Many-worlds interpretation
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The many-worlds interpretation is an interpretation of quantum mechanics that asserts the objective reality of the universal wavefunction and denies the actuality of wavefunction collapse. Many-worlds implies that all possible alternate histories and futures are real, the theory is also referred to as MWI, the relative state formulation, the Everett interpretation, the theory of the universal wavefunction, many-universes interpretation, or just many-worlds. The original relative state formulation is due to Hugh Everett in 1957, later, this formulation was popularized and renamed many-worlds by Bryce Seligman DeWitt in the 1960s and 1970s. The decoherence approaches to interpreting quantum theory have been explored and developed. MWI is one of many multiverse hypotheses in physics and philosophy and it is currently considered a mainstream interpretation along with the other decoherence interpretations, collapse theories, and hidden variable theories such as the Bohmian mechanics. Before many-worlds, reality had always viewed as a single unfolding history. Many-worlds, however, views reality as a tree, wherein every possible quantum outcome is realised. Many-worlds reconciles the observation of events, such as random radioactive decay. In Dublin in 1952 Erwin Schrödinger gave a lecture in which at one point he jocularly warned his audience that what he was about to say might seem lunatic. He went on to assert that when his Nobel equations seem to be describing several different histories, they are not alternatives and this is the earliest known reference to the many-worlds. The idea of MWI originated in Everetts Princeton Ph. D, the phrase many-worlds is due to Bryce DeWitt, who was responsible for the wider popularisation of Everetts theory, which had been largely ignored for the first decade after publication. Under scrutiny of the environment, only pointer states remain unchanged, other states decohere into mixtures of stable pointer states that can persist, and, in this sense, exist, They are einselected. These ideas complement MWI and bring the interpretation in line with our perception of reality, Deutsch is dismissive that many-worlds is an interpretation, saying that calling it an interpretation is like talking about dinosaurs as an interpretation of fossil records. As with the interpretations of quantum mechanics, the many-worlds interpretation is motivated by behavior that can be illustrated by the double-slit experiment. When particles of light are passed through the slit, a calculation assuming wave-like behavior of light can be used to identify where the particles are likely to be observed. Yet when the particles are observed in experiment, they appear as particles. Everetts Ph. D. work provided such an alternative interpretation. e, the state of the observer and the observed are correlated after the observation is made. This led Everett to derive from the unitary, deterministic dynamics alone the notion of a relativity of states, thus the appearance of the objects wavefunctions collapse has emerged from the unitary, deterministic theory itself
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Relational quantum mechanics
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This article is intended for those already familiar with quantum mechanics and its attendant interpretational difficulties. Readers who are new to the subject may first want to read the introduction to quantum mechanics and this interpretation was first delineated by Carlo Rovelli in a 1994 preprint, and has since been expanded upon by a number of theorists. It is inspired by the key idea behind Special Relativity, that the details of an observation depend on the frame of the observer. The physical content of the theory is not to do with objects themselves, as Rovelli puts it, Quantum mechanics is a theory about the physical description of physical systems relative to other systems, and this is a complete description of the world. The state vector of conventional quantum mechanics becomes a description of the correlation of some degrees of freedom in the observer, however, it is held by RQM that this applies to all physical objects, whether or not they are conscious or macroscopic. Any measurement event is simply as an ordinary physical interaction. This incorrect assumption, he said, was that of an observer-independent state of a system, david Mermin has contributed to the relational approach in his Ithaca interpretation. The moniker Zero Worlds has been popularized by Garret to contrast with the Many Worlds Interpretation and this problem was initially discussed in detail in Everetts thesis, The Theory of the Universal Wavefunction. Consider observer O, measuring the state of the quantum system S and we assume that O has complete information on the system, and that O can write down the wavefunction | ψ ⟩ describing it. At the same time, there is another observer O ′, who is interested in the state of the entire O - S system, for our purposes here, we can assume that in a single experiment, the outcome is the eigenstate | ↑ ⟩. So, we may represent the sequence of events in this experiment, with observer O doing the observing, as follows and this is observer O s description of the measurement event. Now, any measurement is also an interaction between two or more systems. According to O, at t 2, the system S is in a determinate state, and, if quantum mechanics is complete, then so is his description. But, for O ′, S is not uniquely determinate, but, if quantum mechanics is complete, then the description that O ′ gives is also complete. Thus the standard formulation of quantum mechanics allows different observers to give different accounts of the same sequence of events. There are many ways to overcome this perceived difficulty, what makes O s description better than that of O ′, or vice versa. Alternatively, we could claim that quantum mechanics is not a theory. Yet another option is to give a preferred status to an observer or type of observer
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Scale relativity
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Scale relativity is a geometrical and fractal space-time theory. The idea of a fractal space-time theory was first introduced by Garnet Ord, the proposal to combine fractal space-time theory with relativity principles was made by Nottale. The resulting scale relativity theory is an extension of the concept of relativity found in special relativity, in physics, relativity theories have shown that position, orientation, movement and acceleration cannot be defined in an absolute way, but only relative to a system of reference. Noticing the relativity of scales, as noticing the other forms of relativity is just a first step, scale relativity theory proposes to extend this insight by introducing an explicit state of scale in coordinate systems. To describe scale transformations requires the use of fractal geometries, which are concerned with scale changes. Scale relativity is thus an extension of relativity theory to the concept of scale, the construction of the theory is similar to previous relativity theories, with three different levels, Galilean, special and general. The development of a general scale relativity is not finished yet. Richard Feynman developed a path integral formulation of quantum mechanics before 1966, searching for the most important paths relevant for quantum particles, Feynman noticed that such paths were very irregular on small scales, i. e. infinite and non-differentiable. This means that in two points, a particle can have not one path, but an infinity of potential paths. This can be illustrated with a concrete example, imagine that you are hiking in the mountains, and that you are free to walk wherever you like. To go from point A to point B, there is not just one path, scale relativity hypothesizes that quantum behavior comes from the fractal nature of spacetime. Indeed, fractal geometries allow to study such non-differentiable paths and this fractal interpretation of quantum mechanics has been further specified by Abbot and Wise, showing that the paths have a fractal dimension 2. Scale relativity goes one further by asserting that the fractality of these paths is a consequence of the fractality of space-time. There are other pioneers who saw the nature of quantum mechanical paths. Garnet Ord and Laurent Nottale both connected fractal space-time with quantum mechanics, Nottale coined the term scale relativity in 1992. He developed the theory and its applications more than one hundred scientific papers. The principle of relativity says that physical laws should be valid in all coordinate systems and this principle has been applied to states of position, as well as to the states of movement of coordinate systems. Such states are never defined in a manner, but relatively to one another
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Quantum field theory
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QFT treats particles as excited states of the underlying physical field, so these are called field quanta. In quantum field theory, quantum mechanical interactions among particles are described by interaction terms among the corresponding underlying quantum fields and these interactions are conveniently visualized by Feynman diagrams, which are a formal tool of relativistically covariant perturbation theory, serving to evaluate particle processes. The first achievement of quantum theory, namely quantum electrodynamics, is still the paradigmatic example of a successful quantum field theory. Ordinarily, quantum mechanics cannot give an account of photons which constitute the prime case of relativistic particles, since photons have rest mass zero, and correspondingly travel in the vacuum at the speed c, a non-relativistic theory such as ordinary QM cannot give even an approximate description. Photons are implicit in the emission and absorption processes which have to be postulated, for instance, the formalism of QFT is needed for an explicit description of photons. In fact most topics in the development of quantum theory were related to the interaction of radiation and matter. However, quantum mechanics as formulated by Dirac, Heisenberg, and Schrödinger in 1926–27 started from atomic spectra, as soon as the conceptual framework of quantum mechanics was developed, a small group of theoreticians tried to extend quantum methods to electromagnetic fields. A good example is the paper by Born, Jordan & Heisenberg. The basic idea was that in QFT the electromagnetic field should be represented by matrices in the way that position. The ideas of QM were thus extended to systems having a number of degrees of freedom. The inception of QFT is usually considered to be Diracs famous 1927 paper on The quantum theory of the emission and absorption of radiation, here Dirac coined the name quantum electrodynamics for the part of QFT that was developed first. Employing the theory of the harmonic oscillator, Dirac gave a theoretical description of how photons appear in the quantization of the electromagnetic radiation field. Later, Diracs procedure became a model for the quantization of fields as well. These first approaches to QFT were further developed during the three years. P. Jordan introduced creation and annihilation operators for fields obeying Fermi–Dirac statistics and these differ from the corresponding operators for Bose–Einstein statistics in that the former satisfy anti-commutation relations while the latter satisfy commutation relations. The methods of QFT could be applied to derive equations resulting from the treatment of particles, e. g. the Dirac equation, the Klein–Gordon equation. Schweber points out that the idea and procedure of second quantization goes back to Jordan, in a number of papers from 1927, some difficult problems concerning commutation relations, statistics, and Lorentz invariance were eventually solved. The first comprehensive account of a theory of quantum fields, in particular
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Quantum information science
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Quantum information science is an area of study based on the idea that information science depends on quantum effects in physics. It includes theoretical issues in computational models as well as more experimental topics in physics including what can. The term quantum information theory is used, but it fails to encompass experimental research in the area. No-communication theorem Quantum decision tree complexity Quantum capacity Quantum communication channel Entanglement-assisted classical capacity Nielsen, M. A. and Chuang, ISBN 0-521-63235-8 Quantiki – quantum information science portal and wiki. ERA-Pilot QIST WP1 European roadmap on Quantum Information Processing and Communication QIIC – Quantum Information, QIP – Quantum Information Group, University of Leeds. The quantum information group at the University of Leeds is engaged in researching a wide spectrum of aspects of quantum information and this ranges from algorithms, quantum computation, to physical implementations of information processing and fundamental issues in quantum mechanics. Also contains some basic tutorials for the lay audience, mathQI Research Group on Mathematics and Quantum Information. CQT Centre for Quantum Technologies at the National University of Singapore
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Quantum computing
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Quantum computing studies theoretical computation systems that make direct use of quantum-mechanical phenomena, such as superposition and entanglement, to perform operations on data. Quantum computers are different from binary digital electronic computers based on transistors, a quantum Turing machine is a theoretical model of such a computer, and is also known as the universal quantum computer. The field of computing was initiated by the work of Paul Benioff and Yuri Manin in 1980, Richard Feynman in 1982. A quantum computer with spins as quantum bits was also formulated for use as a quantum space–time in 1968, there exist quantum algorithms, such as Simons algorithm, that run faster than any possible probabilistic classical algorithm. A classical computer could in principle simulate a quantum algorithm, as quantum computation does not violate the Church–Turing thesis, on the other hand, quantum computers may be able to efficiently solve problems which are not practically feasible on classical computers. A classical computer has a made up of bits, where each bit is represented by either a one or a zero. A quantum computer maintains a sequence of qubits, in general, a quantum computer with n qubits can be in an arbitrary superposition of up to 2 n different states simultaneously. A quantum computer operates by setting the qubits in a drift that represents the problem at hand. The sequence of gates to be applied is called a quantum algorithm, the calculation ends with a measurement, collapsing the system of qubits into one of the 2 n pure states, where each qubit is zero or one, decomposing into a classical state. The outcome can therefore be at most n classical bits of information, Quantum algorithms are often probabilistic, in that they provide the correct solution only with a certain known probability. Note that the term non-deterministic computing must not be used in case to mean probabilistic. An example of an implementation of qubits of a computer could start with the use of particles with two spin states, down and up. This is true because any such system can be mapped onto an effective spin-1/2 system, a quantum computer with a given number of qubits is fundamentally different from a classical computer composed of the same number of classical bits. This means that when the state of the qubits is measured. To better understand this point, consider a classical computer that operates on a three-bit register, if there is no uncertainty over its state, then it is in exactly one of these states with probability 1. However, if it is a computer, then there is a possibility of it being in any one of a number of different states. The state of a quantum computer is similarly described by an eight-dimensional vector. Here, however, the coefficients a k are complex numbers, and it is the sum of the squares of the absolute values, ∑ i | a i |2
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Quantum chaos
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Quantum chaos is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is, What is the relationship between quantum mechanics and classical chaos, the correspondence principle states that classical mechanics is the classical limit of quantum mechanics. If this is true, then there must be quantum mechanisms underlying classical chaos, correlating statistical descriptions of eigenvalues with the classical behavior of the same Hamiltonian. Semiclassical methods such as periodic-orbit theory connecting the classical trajectories of the system with quantum features. Direct application of the correspondence principle, during the first half of the twentieth century, chaotic behavior in mechanics was recognized, but not well-understood. The foundations of quantum mechanics were laid in that period. Other phenomena show up in the evolution of a quantum system. In some contexts, such as acoustics or microwaves, wave patterns are directly observable, Quantum chaos typically deals with systems whose properties need to be calculated using either numerical techniques or approximation schemes. Simple and exact solutions are precluded by the fact that the systems constituents either influence each other in a complex way, finding constants of motion so that this separation can be performed can be a difficult analytical task. Solving the classical problem can give insight into solving the quantum problem. If there are regular classical solutions of the same Hamiltonian, then there are constants of motion. Other approaches have developed in recent years. One is to express the Hamiltonian in different coordinate systems in different regions of space, wavefunctions are obtained in these regions, and eigenvalues are obtained by matching boundary conditions. Another approach is numerical matrix diagonalization, if the Hamiltonian matrix is computed in any complete basis, eigenvalues and eigenvectors are obtained by diagonalizing the matrix. However, all complete basis sets are infinite, and we need to truncate the basis and these techniques boil down to choosing a truncated basis from which accurate wavefunctions can be constructed. A given Hamiltonian shares the same constants of motion for both classical and quantum dynamics, Quantum systems can also have additional quantum numbers corresponding to discrete symmetries. Nevertheless, learning how to solve such problems is an important part of answering the question of quantum chaos. Statistical measures of quantum chaos were born out of a desire to quantify spectral features of complex systems, random matrix theory was developed in an attempt to characterize spectra of complex nuclei
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Density matrix
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A density matrix is a matrix that describes a quantum system in a mixed state, a statistical ensemble of several quantum states. This should be contrasted with a state vector that describes a quantum system in a pure state. The density matrix is the analogue to a phase-space probability measure in classical statistical mechanics. Mixed states arise in situations where the experimenter does not know which particular states are being manipulated, examples include a system in thermal equilibrium or a system with an uncertain or randomly varying preparation history. Also, if a system has two or more subsystems that are entangled, then each subsystem must be treated as a mixed state even if the complete system is in a pure state. The density matrix is also a tool in quantum decoherence theory. The density matrix is a representation of an operator called the density operator. The density matrix is obtained from the density operator by choice of basis in the underlying space, in practice, the terms density matrix and density operator are often used interchangeably. Both matrix and operator are self-adjoint, positive semi-definite, of trace one, the formalism of density operators and matrices was introduced by John von Neumann in 1927 and independently, but less systematically by Lev Landau and Felix Bloch in 1927 and 1946 respectively. In quantum mechanics, the state of a system is represented by a state vector | ψ ⟩. A quantum system with a state vector | ψ ⟩ is called a pure state and this system would be in a mixed state. The density matrix is useful for mixed states, because any state, pure or mixed. A mixed state is different from a quantum superposition, the probabilities in a mixed state are classical probabilities, unlike the quantum probabilites in a quantum superposition. In fact, a superposition of pure states is another pure state. In this case, the coefficients 1 /2 are not probabilities, an example of pure and mixed states is light polarization. Photons can have two helicities, corresponding to two quantum states, | R ⟩ and | L ⟩. A photon can also be in a state, such as /2 or /2. More generally, it can be in any state α | R ⟩ + β | L ⟩, corresponding to linear, circular, or elliptical polarization
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Scattering theory
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In mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Wave scattering corresponds to the collision and scattering of a wave with some material object, the direct scattering problem is the problem of determining the distribution of scattered radiation/particle flux basing on the characteristics of the scatterer. The inverse scattering problem is the problem of determining the characteristics of an object from measurement data of radiation or particles scattered from the object, the concepts used in scattering theory go by different names in different fields. The object of section is to point the reader to common threads. e. that d I d x = − Q I where Q is an interaction coefficient. Hence one converts between these quantities via Q = 1/λ = ησ = ρ/τ, as shown in the figure at left, in electromagnetic absorption spectroscopy, for example, interaction coefficient is variously called opacity, absorption coefficient, and attenuation coefficient. In mathematical physics, scattering theory is a framework for studying and understanding the interaction or scattering of solutions to differential equations. In acoustics, the equation is the wave equation, and scattering studies how its solutions. In the case of classical electrodynamics, the equation is again the wave equation. In particle physics, the equations are those of Quantum electrodynamics, Quantum chromodynamics and the Standard Model, the solutions of interest describe the long-term motion of free atoms, molecules, photons, electrons, and protons. The scenario is that several particles come together from a distance away. These reagents then collide, optionally reacting, getting destroyed or creating new particles, the products and unused reagents then fly away to infinity again. The solutions reveal which directions the products are most likely to fly off to and they also reveal the probability of various reactions, creations, and decays occurring. There are two predominant techniques of finding solutions to scattering problems, partial wave analysis, and the Born approximation, the term elastic scattering implies that the internal states of the scattering particles do not change, and hence they emerge unchanged from the scattering process. The example of scattering in quantum chemistry is particularly instructive, as the theory is reasonably complex while still having a foundation on which to build an intuitive understanding. When two atoms are scattered off one another, one can understand them as being the bound state solutions of differential equation. Thus, for example, the hydrogen atom corresponds to a solution to the Schrödinger equation with a negative inverse-power central potential. The scattering of two hydrogen atoms will disturb the state of each atom, resulting in one or both becoming excited, or even ionized, representing an inelastic scattering process, the term deep inelastic scattering refers to a special kind of scattering experiment in particle physics. In mathematics, scattering theory deals with an abstract formulation of the same set of concepts
37.
Yakir Aharonov
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Yakir Aharonov is an Israeli physicist specializing in quantum physics. He is a Professor of Theoretical Physics and the James J. Farley Professor of Natural Philosophy at Chapman University in California and he is also a distinguished professor in the Perimeter Institute and a professor emeritus at Tel Aviv University in Israel. He is president of the IYAR, The Israeli Institute for Advanced Research, yakir Aharonov was born in Haifa. He received his education at the Technion – Israel Institute of Technology in Haifa. He continued his studies at the Technion and then moved to Bristol University, UK together with his doctoral advisor David Bohm. His research interests are nonlocal and topological effects in quantum mechanics, quantum field theories, in 1959, he and David Bohm proposed the Aharonov–Bohm effect for which he co-received the 1998 Wolf Prize. In 1988 Aharonov et al. published their theory of weak values and this work was motivated by Aharonovs long time quest to experimentally verify his theory that apparently random events in quantum mechanics are caused by events in the future. Verifying a present effect of a future cause requires a measurement and he and his colleagues claim that they were able to use weak measurements and verify the present effect of the future cause. 2008–Present, Professor of Theoretical Physics and the James J.2010, National Medal of Science, awarded and presented by President Barack Obama
38.
John Stewart Bell
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John Stewart Bell FRS was a Northern Irish physicist, and the originator of Bells theorem, an important theorem in quantum physics regarding hidden variable theories. John Bell was born in Belfast, Northern Ireland, both sides of his family were of Ulster Scots roots. When he was 11 years old, he decided to be a scientist, Bell then attended the Queens University of Belfast, and obtained a bachelors degree in experimental physics in 1948, and one in mathematical physics a year later. He went on to complete a Ph. D. in physics at the University of Birmingham in 1956, specialising in nuclear physics, in 1954, he married Mary Ross, also a physicist, whom he had met while working on accelerator physics at Malvern, UK. Bell became a vegetarian in his teen years, according to his wife, Bell was an atheist. Bells career began with the UK Atomic Energy Research Establishment, near Harwell, Oxfordshire, after several years he moved to work for the European Organization for Nuclear Research, in Geneva, Switzerland. There he worked almost exclusively on theoretical physics and on accelerator design. He was elected a Foreign Honorary Member of the American Academy of Arts, also of significance during his career, Bell, together with John Bradbury Sykes, M. J. Kearsley, and W. H. Bell was a proponent of pilot wave theory. In this work, he showed that carrying forward EPRs analysis permits one to derive the famous Bells theorem, the resultant inequality, derived from certain assumptions, is violated by quantum theory. There is some disagreement regarding what Bells inequality—in conjunction with the EPR analysis—can be said to imply, according to an alternative interpretation, not all local theories in general, but only local hidden variables theories have shown to be incompatible with the predictions of quantum theory. More plausible to me is that we find that there is no boundary. The wave functions would prove to be a provisional or incomplete description of the quantum-mechanical part and it is this possibility, of a homogeneous account of the world, which is for me the chief motivation of the study of the so-called hidden variable possibility. Bell was impressed that in the formulation of David Bohm’s nonlocal hidden variable theory, no such boundary is needed, and it was this which sparked his interest in the field of research. Bell also criticized the standard formalism of quantum mechanics on the grounds of lack of physical precision and this is clear already from their vocabulary. On this list of bad words from good books, the worst of all is measurement, but if he were to thoroughly explore the viability of Bohms theory, Bell needed to answer the challenge of the so-called impossibility proofs against hidden variables. Bell addressed these in a paper entitled On the Problem of Hidden Variables in Quantum Mechanics, Bell subsequently claimed, The proof of von Neumann is not merely false but foolish. In this same work, Bell showed that an effort at such a proof also fails to eliminate the hidden variables program. The supposed flaw in von Neumanns proof had been discovered by Grete Hermann in 1935
39.
Patrick Blackett
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He also made a major contribution in World War II advising on military strategy and developing operational research. His left-wing views saw an outlet in third world development and in influencing policy in the Labour Government of the 1960s, Blackett was born in Kensington, London, the son of Arthur Stuart Blackett, a stockbroker, and his wife Caroline Maynard. His younger sister was the psychoanalyst Marion Milner and his paternal grandfather Rev. Henry Blackett, brother of Edmund Blacket the Australian architect, was for many years Vicar of Croydon. His maternal grandfather Charles Maynard was an officer in the Royal Artillery at the time of the Indian Mutiny, the Blackett family lived successively at Kensington, Kenley, Woking and Guildford, Surrey, where Blackett went to preparatory school. His main hobbies were model aeroplanes and crystal radio and he was accepted and spent two years there before moving on to Dartmouth where he was usually head of his class. In August 1914 on the outbreak of World War I Blackett was assigned to service as a midshipman. He was transferred to the Cape Verde Islands on HMS Carnarvon and was present at the Battle of the Falkland Islands and he was then transferred to HMS Barham and saw much action at the Battle of Jutland. While on HMS Barham, Blackett was co-inventor of a device on which the Admiralty took out a patent. In 1916 he applied to join the RNAS but his application was refused, in October that year he became a sub-lieutenant on HMS P17 on Dover patrol, and in July 1917 he was posted to HMS Sturgeon in the Harwich Force under Admiral Tyrwhitt. Blackett was particularly concerned by the quality of gunnery in the force compared with that of the enemy and of his own previous experience. He was promoted to Lieutenant in May 1918, but had decided to leave the Navy, then, in January 1919, the Admiralty sent the officers whose training had been interrupted by the war to Cambridge University for a course of general duties. On his first night at Magdalene College, Cambridge he met Kingsley Martin and Geoffrey Webb, later recalling that he had never before, in his naval training, Blackett was impressed by the prestigious Cavendish Laboratory, and left the Navy to study mathematics and physics at Cambridge. Rutherford had found out that the nucleus of the nitrogen atom could be disintegrated by firing fast alpha particles into nitrogen. He asked Blackett to use a chamber to find visible tracks of this disintegration. Eight of these were forked, and this showed that the nitrogen atom-alpha particle combination had formed an atom of fluorine, which then disintegrated into an isotope of oxygen, Blackett spent some time in 1924–1925 at Göttingen, Germany working with James Franck on atomic spectra. In 1932, working with Giuseppe Occhialini, he devised a system of geiger counters which only took photographs when a ray particle traversed the chamber. They found 500 tracks of high energy cosmic ray particles in 700 automatic exposures, in 1933, Blackett discovered fourteen tracks which confirmed the existence of the positron and revealed the now instantly recognisable opposing spiral traces of positron/electron pair production. This work and that on annihilation radiation made him one of the first and that same year he moved to Birkbeck College, University of London as Professor of Physics for four years
40.
Felix Bloch
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Felix Bloch was a Swiss physicist, working mainly in the U. S. He and Edward Mills Purcell were awarded the 1952 Nobel Prize for their development of new ways, in 1954–1955, he served for one year as the first Director-General of CERN. Bloch was born in Zürich, Switzerland to Jewish parents Gustav and he was educated at the Cantonal Gymnasium in Zurich and at the Eidgenössische Technische Hochschule, also in Zürich. Initially studying engineering he soon changed to physics, during this time he attended lectures and seminars given by Peter Debye and Hermann Weyl at ETH Zürich and Erwin Schrödinger at the neighboring University of Zürich. A fellow student in these seminars was John von Neumann, graduating in 1927 he continued his physics studies at the University of Leipzig with Werner Heisenberg, gaining his doctorate in 1928. His doctoral thesis established the theory of solids, using Bloch waves to describe the electrons. In 1940 he married Lore Misch and he remained in European academia, studying with Wolfgang Pauli in Zürich, Niels Bohr in Copenhagen and Enrico Fermi in Rome before he went back to Leipzig assuming a position as privatdozent. In 1933, immediately after Hitler came to power, he left Germany because he was Jewish and he emigrated to work at Stanford University in 1934. In the fall of 1938, Bloch began working with the 37 cyclotron at the University of California at Berkeley to determine the moment of the neutron. Bloch went on to become the first professor for physics at Stanford. In 1939, he became a citizen of the United States. During WW II he worked on power at Los Alamos National Laboratory. After the war he concentrated on investigations into nuclear induction and nuclear magnetic resonance, in 1946 he proposed the Bloch equations which determine the time evolution of nuclear magnetization. After leaving CERN, he returned to Stanford University, where he in 1961 was made Max Stein Professor of Physics, at Stanford, he was the advisor of Carson D. Jeffries, who became a professor of Physics at the University of California, Berkeley. In 1964, he was elected a member of the Royal Netherlands Academy of Arts. List of Jewish Nobel laureates List of things named after Felix Bloch Nobel Prize for Physics,1952, james T. White & Co.1984. Fission Spectrum, Los Alamos National Laboratory, United States Department of Energy. org/physics/laureates/1952/bloch-bio. html http, //www-sul. stanford. edu/depts/spc/xml/sc0303
41.
David Bohm
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To complement it, he developed a mathematical and physical theory of implicate and explicate order. In this, his epistemology mirrored his ontology, due to his Communist affiliations, Bohm was the subject of a federal government investigation in 1949, prompting him to leave the United States. He pursued his career in several countries, becoming first a Brazilian. Bohm was born in Wilkes-Barre, Pennsylvania, United States, to a Hungarian Jewish immigrant father, Samuel Bohm, and he was raised mainly by his father, a furniture-store owner and assistant of the local rabbi. Despite being raised in a Jewish family, he became an agnostic in his teenage years, Bohm attended Pennsylvania State College, graduating in 1939, and then the California Institute of Technology, for one year. He then transferred to the physics group directed by Robert Oppenheimer at the University of California, Berkeley. Bohm lived in the neighborhood as some of Oppenheimers other graduate students. He was active in Communist and Communist-backed organizations, including the Young Communist League, the Campus Committee to Fight Conscription, during World War II, the Manhattan Project mobilized much of Berkeleys physics research in the effort to produce the first atomic bomb. Though Oppenheimer had asked Bohm to work with him at Los Alamos, Bohm remained in Berkeley, teaching physics, until he completed his Ph. D. in 1943 by an unusual circumstance. According to Peat, the calculations that he had completed proved useful to the Manhattan Project and were immediately classified. Without security clearance, Bohm was denied access to his own work, not only would he be barred from defending his thesis, to satisfy the university, Oppenheimer certified that Bohm had successfully completed the research. Bohm later performed theoretical calculations for the Calutrons at the Y-12 facility in Oak Ridge, after the war, Bohm became an assistant professor at Princeton University, where he worked closely with Albert Einstein. In May 1949, the House Un-American Activities Committee called upon Bohm to testify because of his previous ties to suspected Communists, Bohm invoked his Fifth amendment right to refuse to testify, and refused to give evidence against his colleagues. In 1950 Bohm was arrested for refusing to answer HUACs questions and he was acquitted in May 1951, but Princeton had already suspended him. His request to go to Manchester received Einsteins support but was unsuccessful, Bohm then left for Brazil to assume a professorship of physics at the University of São Paulo at Jayme Tiomnos invitation, and on Einsteins and Oppenheimers recommendation. During his early period, Bohm made a number of significant contributions to physics, particularly quantum mechanics, as a post-graduate at Berkeley, he developed a theory of plasmas, discovering the electron phenomenon known now as Bohm-diffusion. His first book, Quantum Theory, published in 1951, was received by Einstein. But Bohm became dissatisfied with the interpretation of quantum theory he had written about in that book
42.
Niels Bohr
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Niels Henrik David Bohr was a Danish physicist who made foundational contributions to understanding atomic structure and quantum theory, for which he received the Nobel Prize in Physics in 1922. Bohr was also a philosopher and a promoter of scientific research, although the Bohr model has been supplanted by other models, its underlying principles remain valid. He conceived the principle of complementarity, that items could be analysed in terms of contradictory properties. The notion of complementarity dominated Bohrs thinking in science and philosophy. Bohr founded the Institute of Theoretical Physics at the University of Copenhagen, now known as the Niels Bohr Institute, Bohr mentored and collaborated with physicists including Hans Kramers, Oskar Klein, George de Hevesy, and Werner Heisenberg. He predicted the existence of a new element, which was named hafnium, after the Latin name for Copenhagen. Later, the element bohrium was named after him, during the 1930s, Bohr helped refugees from Nazism. After Denmark was occupied by the Germans, he had a meeting with Heisenberg. In September 1943, word reached Bohr that he was about to be arrested by the Germans, from there, he was flown to Britain, where he joined the British Tube Alloys nuclear weapons project, and was part of the British mission to the Manhattan Project. After the war, Bohr called for cooperation on nuclear energy. He had a sister, Jenny, and a younger brother Harald. Jenny became a teacher, while Harald became a mathematician and Olympic footballer who played for the Danish national team at the 1908 Summer Olympics in London. Bohr was a footballer as well, and the two brothers played several matches for the Copenhagen-based Akademisk Boldklub, with Bohr as goalkeeper. Bohr was educated at Gammelholm Latin School, starting when he was seven, in 1903, Bohr enrolled as an undergraduate at Copenhagen University. His major was physics, which he studied under Professor Christian Christiansen and he also studied astronomy and mathematics under Professor Thorvald Thiele, and philosophy under Professor Harald Høffding, a friend of his father. This involved measuring the frequency of oscillation of the radius of a water jet, Bohr conducted a series of experiments using his fathers laboratory in the university, the university itself had no physics laboratory. To complete his experiments, he had to make his own glassware and his essay, which he submitted at the last minute, won the prize. He later submitted a version of the paper to the Royal Society in London for publication in the Philosophical Transactions of the Royal Society
43.
Max Born
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Max Born was a German physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to physics and optics and supervised the work of a number of notable physicists in the 1920s and 1930s. Born won the 1954 Nobel Prize in Physics for his research in Quantum Mechanics. He wrote his Ph. D. thesis on the subject of Stability of Elastica in a Plane and Space, in 1905, he began researching special relativity with Minkowski, and subsequently wrote his habilitation thesis on the Thomson model of the atom. In the First World War, after originally being placed as a radio operator, in 1921, Born returned to Göttingen, arranging another chair for his long-time friend and colleague James Franck. Under Born, Göttingen became one of the worlds foremost centres for physics, in 1925, Born and Werner Heisenberg formulated the matrix mechanics representation of quantum mechanics. The following year, he formulated the now-standard interpretation of the probability density function for ψ*ψ in the Schrödinger equation and his influence extended far beyond his own research. Max Delbrück, Siegfried Flügge, Friedrich Hund, Pascual Jordan, Maria Goeppert-Mayer, Lothar Wolfgang Nordheim, Robert Oppenheimer, in January 1933, the Nazi Party came to power in Germany, and Born, who was Jewish, was suspended. Max Born became a naturalised British subject on 31 August 1939 and he remained at Edinburgh until 1952. He retired to Bad Pyrmont, in West Germany, and died in a hospital in Göttingen on 5 January 1970. Max Born was born on 11 December 1882 in Breslau, which at the time of Borns birth was part of the Prussian Province of Silesia in the German Empire and she died when Max was four years old, on 29 August 1886. Max had a sister, Käthe, who was born in 1884, Wolfgang later became Professor of Art History at the City College of New York. Initially educated at the König-Wilhelm-Gymnasium in Breslau, Born entered the University of Breslau in 1901, the German university system allowed students to move easily from one university to another, so he spent summer semesters at Heidelberg University in 1902 and the University of Zurich in 1903. Fellow students at Breslau, Otto Toeplitz and Ernst Hellinger, told Born about the University of Göttingen, at Göttingen he found three renowned mathematicians, Felix Klein, David Hilbert and Hermann Minkowski. Very soon after his arrival, Born formed close ties to the two men. Being class scribe put Born into regular, invaluable contact with Hilbert, Hilbert became Borns mentor after selecting him to be the first to hold the unpaid, semi-official position of assistant. Borns introduction to Minkowski came through Borns stepmother, Bertha, as she knew Minkowski from dancing classes in Königsberg, the introduction netted Born invitations to the Minkowski household for Sunday dinners. In addition, while performing his duties as scribe and assistant, Borns relationship with Klein was more problematic
44.
Satyendra Nath Bose
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Satyendra Nath Bose, FRS was an Indian physicist from Bengal specialising in theoretical physics. He is best known for his work on mechanics in the early 1920s, providing the foundation for Bose–Einstein statistics. A Fellow of the Royal Society, he was awarded Indias second highest civilian award, the class of particles that obey Bose–Einstein statistics, bosons, was named after Bose by Paul Dirac. A self-taught scholar and a polymath, he had a range of interests in varied fields including physics, mathematics, chemistry, biology, mineralogy, philosophy, arts, literature. He served on many research and development committees in sovereign India, Bose was born in Calcutta, the eldest of seven children. He was the son, with six sisters after him. His ancestral home was in village Bara Jagulia, in the district of Nadia and his schooling began at the age of five, near his home. When his family moved to Goabagan, he was admitted to the New Indian School, in the final year of school, he was admitted to the Hindu School. He passed his examination in 1909 and stood fifth in the order of merit. Naman Sharma and Meghnad Saha, from Dacca, joined the college two years later. Prasanta Chandra Mahalanobis and Sisir Kumar Mitra were few years senior to Bose, Bose chose mixed mathematics for his BSc and passed the examinations standing first in 1913 and again stood first in the MSc mixed mathematics exam in 1915. It is said that his marks in the MSc examination created a new record in the annals of the University of Calcutta, after completing his MSc, Bose joined the University of Calcutta as a research scholar in 1916 and started his studies in the theory of relativity. It was an era in the history of scientific progress. Quantum theory had just appeared on the horizon and important results had started pouring in and his father, Surendranath Bose, worked in the Engineering Department of the East Indian Railway Company. In 1914, at age 20, Satyendra Nath Bose married Ushabati Ghosh and they had nine children, two of whom died in early childhood. When he died in 1974, he left behind his wife, as a polyglot, Bose was well versed in several languages such as Bengali, English, French, German and Sanskrit as well as the poetry of Lord Tennyson, Rabindranath Tagore and Kalidasa. He could play the esraj, an instrument similar to a violin. He was actively involved in running night schools that came to be known as the Working Mens Institute and he came in contact with teachers such as Jagadish Chandra Bose, Prafulla Chandra Ray and Naman Sharma who provided inspiration to aim high in life
45.
Louis de Broglie
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Louis-Victor-Pierre-Raymond, 7e duc de Broglie was a French physicist who made groundbreaking contributions to quantum theory. In his 1924 PhD thesis he postulated the wave nature of electrons and this concept is known as the de Broglie hypothesis, an example of wave–particle duality, and forms a central part of the theory of quantum mechanics. De Broglie won the Nobel Prize for Physics in 1929, after the behaviour of matter was first experimentally demonstrated in 1927. The 1925 pilot-wave model, and the behaviour of particles discovered by de Broglie was used by Erwin Schrödinger in his formulation of wave mechanics. The pilot-wave model and interpretation was abandoned, in favor of the quantum formalism. Louis de Broglie was the sixteenth member elected to occupy seat 1 of the Académie française in 1944, De Broglie became the first high-level scientist to call for establishment of a multi-national laboratory, a proposal that led to the establishment of the European Organization for Nuclear Research. Louis de Broglie was born to a family in Dieppe, Seine-Maritime, younger son of Victor. He became the 7th duc de Broglie in 1960 upon the death without heir of his brother, Maurice, 6th duc de Broglie. When he died in Louveciennes, he was succeeded as duke by a distant cousin, Victor-François, De Broglie had intended a career in humanities, and received his first degree in history. Afterwards, though, he turned his attention toward mathematics and physics, with the outbreak of the First World War in 1914, he offered his services to the army in the development of radio communications. His 1924 thesis Recherches sur la théorie des quanta introduced his theory of electron waves and this included the wave–particle duality theory of matter, based on the work of Max Planck and Albert Einstein on light. This research culminated in the de Broglie hypothesis stating that any moving particle or object had an associated wave, De Broglie thus created a new field in physics, the mécanique ondulatoire, or wave mechanics, uniting the physics of energy and matter. For this he won the Nobel Prize in Physics in 1929, the theory has since been known as the De Broglie–Bohm theory. In addition to scientific work, de Broglie thought and wrote about the philosophy of science. De Broglie became a member of the Académie des sciences in 1933 and he was asked to join Le Conseil de lUnion Catholique des Scientifiques Francais, but declined because he was non-religious and an atheist. On 12 October 1944, he was elected to the Académie française, in an event unique in the history of the Académie, he was received as a member by his own brother Maurice, who had been elected in 1934. UNESCO awarded him the first Kalinga Prize in 1952 for his work in popularizing scientific knowledge, in 1961 he received the title of Knight of the Grand Cross in the Légion dhonneur. De Broglie was awarded a post as counselor to the French High Commission of Atomic Energy in 1945 for his efforts to bring industry and he established a center for applied mechanics at the Henri Poincaré Institute, where research into optics, cybernetics, and atomic energy were carried out
46.
Arthur Compton
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It was a sensational discovery at the time, the wave nature of light had been well-demonstrated, but the idea that light had both wave and particle properties was not easily accepted. He is also known for his leadership of the Manhattan Projects Metallurgical Laboratory, in 1919, Compton was awarded one of the first two National Research Council Fellowships that allowed students to study abroad. He chose to go to Cambridge Universitys Cavendish Laboratory in England, further research along these lines led to the discovery of the Compton effect. During World War II, Compton was a key figure in the Manhattan Project that developed the first nuclear weapons and his reports were important in launching the project. Compton oversaw Enrico Fermis creation of Chicago Pile-1, the first nuclear reactor, the Metallurgical Laboratory was also responsible for the design and operation of the X-10 Graphite Reactor at Oak Ridge, Tennessee. Plutonium began being produced in the Hanford Site reactors in 1945, after the war, Compton became Chancellor of Washington University in St. Louis. Arthur Compton was born on September 10,1892, in Wooster, Ohio, the son of Elias and Otelia Catherine Compton, Elias was dean of the University of Wooster, which Arthur also attended. Arthurs eldest brother, Karl, who also attended Wooster, earned a PhD in physics from Princeton University in 1912, all three brothers were members of the Alpha Tau Omega fraternity. Compton was initially interested in astronomy, and took a photograph of Halleys Comet in 1910, around 1913, he described an experiment where an examination of the motion of water in a circular tube demonstrated the rotation of the earth. That year, he graduated from Wooster with a Bachelor of Science degree and entered Princeton, where he received his Master of Arts degree in 1914. Compton then studied for his PhD in physics under the supervision of Hereward L. Cooke, writing his dissertation on The intensity of X-ray reflection, and the distribution of the electrons in atoms. When Arthur Compton earned his PhD in 1916, he, Karl, later, they would become the first such trio to simultaneously head American colleges. Their sister Mary married a missionary, C. Herbert Rice, in June 1916, Compton married Betty Charity McCloskey, a Wooster classmate and fellow graduate. They had two sons, Arthur Alan and John Joseph Compton, during World War I he developed aircraft instrumentation for the Signal Corps. In 1919, Compton was awarded one of the first two National Research Council Fellowships that allowed students to study abroad and he chose to go to Cambridge Universitys Cavendish Laboratory in England. Working with George Paget Thomson, the son of J. J. Thomson, Compton studied the scattering and he observed that the scattered rays were more easily absorbed than the original source. Compton was greatly impressed by the Cavendish scientists, especially Ernest Rutherford, Charles Galton Darwin and Arthur Eddington, for a time Compton was a deacon at a Baptist church. Science can have no quarrel, he said, with a religion which postulates a God to whom men are as His children
47.
Paul Dirac
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Paul Adrien Maurice Dirac OM FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics. Among other discoveries, he formulated the Dirac equation which describes the behaviour of fermions, Dirac shared the 1933 Nobel Prize in Physics with Erwin Schrödinger for the discovery of new productive forms of atomic theory. He also made significant contributions to the reconciliation of general relativity with quantum mechanics and he was regarded by his friends and colleagues as unusual in character. Albert Einstein said of him This balancing on the path between genius and madness is awful. He is regarded as one of the most significant physicists of the 20th century, Paul Adrien Maurice Dirac was born at his parents home in Bristol, England, on 8 August 1902, and grew up in the Bishopston area of the city. His father, Charles Adrien Ladislas Dirac, was an immigrant from Saint-Maurice, Switzerland and his mother, Florence Hannah Dirac, née Holten, the daughter of a ships captain, was born in Cornwall, England, and worked as a librarian at the Bristol Central Library. Paul had a sister, Béatrice Isabelle Marguerite, known as Betty, and an older brother, Reginald Charles Félix, known as Felix. Dirac later recalled, My parents were terribly distressed, I didnt know they cared so much I never knew that parents were supposed to care for their children, but from then on I knew. Charles and the children were officially Swiss nationals until they became naturalised on 22 October 1919, Diracs father was strict and authoritarian, although he disapproved of corporal punishment. Dirac had a relationship with his father, so much so that after his fathers death, Dirac wrote, I feel much freer now. Charles forced his children to speak to him only in French, when Dirac found that he could not express what he wanted to say in French, he chose to remain silent. Dirac was educated first at Bishop Road Primary School and then at the all-boys Merchant Venturers Technical College, the school was an institution attached to the University of Bristol, which shared grounds and staff. It emphasised technical subjects like bricklaying, shoemaking and metal work and this was unusual at a time when secondary education in Britain was still dedicated largely to the classics, and something for which Dirac would later express his gratitude. Dirac studied electrical engineering on a City of Bristol University Scholarship at the University of Bristols engineering faculty, shortly before he completed his degree in 1921, he sat the entrance examination for St Johns College, Cambridge. He passed, and was awarded a £70 scholarship, but this short of the amount of money required to live. Instead he took up an offer to study for a Bachelor of Arts degree in mathematics at the University of Bristol free of charge and he was permitted to skip the first year of the course owing to his engineering degree. In 1923, Dirac graduated, once again with first class honours, along with his £70 scholarship from St Johns College, this was enough to live at Cambridge. From 1925 to 1928 he held an 1851 Research Fellowship from the Royal Commission for the Exhibition of 1851 and he completed his PhD in June 1926 with the first thesis on quantum mechanics to be submitted anywhere
48.
Clinton Davisson
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Clinton Joseph Davisson, was an American physicist who won the 1937 Nobel Prize in Physics for his discovery of electron diffraction in the famous Davisson-Germer experiment. Davisson shared the Nobel Prize with George Paget Thomson, who independently discovered electron diffraction at about the time as Davisson. Davisson was born in Bloomington, Illinois and he graduated from Bloomington High School in 1902, and entered the University of Chicago on scholarship. Upon the recommendation of Robert A. Millikan, in 1905 Davisson was hired by Princeton University as Instructor of Physics and he completed the requirements for his B. S. degree from Chicago in 1908, mainly by working in the summers. While teaching at Princeton, he did doctoral research with Owen Richardson. He received his Ph. D. in physics from Princeton in 1911, in the year he married Richardsons sister. Davisson was then appointed as an assistant professor at the Carnegie Institute of Technology, in 1917 he took a leave from the Carnegie Institute to do war-related research with the Engineering Department of the Western Electric Company. At the end of the war, Davisson accepted a permanent position at Western Electric after receiving assurances of his freedom there to do basic research and he had found that his teaching responsibilities at the Carnegie Institute largely precluded him from doing research. Davisson remained at Western Electric until his retirement in 1946. He then accepted a research appointment at the University of Virginia that continued until his second retirement in 1954. In the 19th Century, diffraction was well established for light, in 1927, while working for Bell Labs, Davisson and Lester Germer performed an experiment showing that electrons were diffracted at the surface of a crystal of nickel. This celebrated Davisson-Germer experiment confirmed the de Broglie hypothesis that particles of matter have a wave-like nature, in particular, their observation of diffraction allowed the first measurement of a wavelength for electrons. The measured wavelength λ agreed well with de Broglies equation λ = h / p, while doing his graduate work at Princeton, Davisson met his wife and life companion Charlotte Sara Richardson, who was visiting her brother, Professor Richardson. Richardson is the sister-in-law of Oswald Veblen, a prominent mathematician, clinton and Charlotte Davisson had four children, including the American physicist Richard Davisson. The crater Davisson on the Moon is named after him, Davisson died on February 1,1958, at the age of 76. An impact crater on the far side of the moon was named after Davisson in 1970 by the IAU, nobelprize. org Biography Bloomington native won Nobel Prize in physics - Pantagraph
49.
Peter Debye
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Peter Joseph William Debye ForMemRS was a Dutch-American physicist and physical chemist, and Nobel laureate in Chemistry. Born Petrus Josephus Wilhelmus Debije in Maastricht, Netherlands, Debye enrolled in the Aachen University of Technology in 1901, in 1905, he completed his first degree in electrical engineering. He published his first paper, an elegant solution of a problem involving eddy currents. At Aachen, he studied under the theoretical physicist Arnold Sommerfeld, in 1906, Sommerfeld received an appointment at Munich, Bavaria, and took Debye with him as his assistant. Debye got his Ph. D. with a dissertation on radiation pressure in 1908, in 1910, he derived the Planck radiation formula using a method which Max Planck agreed was simpler than his own. In 1911, when Albert Einstein took an appointment as a professor at Prague, Bohemia, Debye took his old professorship at the University of Zurich and he was awarded the Lorentz Medal in 1935. From 1937 to 1939 he was the president of the Deutsche Physikalische Gesellschaft, in May 1914 he became member of the Royal Netherlands Academy of Arts and Sciences and in December of the same year he became foreign member. In 1913, Debye married Mathilde Alberer and they had a son, Peter P. Debye, and a daughter, Mathilde Maria. Peter became a physicist and collaborated with Debye in some of his researches, in consequence, the units of molecular dipole moments are termed debyes in his honor. Also in 1912, he extended Albert Einsteins theory of heat to lower temperatures by including contributions from low-frequency phonons. In 1913, he extended Niels Bohrs theory of structure, introducing elliptical orbits. In 1914–1915, Debye calculated the effect of temperature on X-ray diffraction patterns of crystalline solids with Paul Scherrer, in 1923, together with his assistant Erich Hückel, he developed an improvement of Svante Arrhenius theory of electrical conductivity in electrolyte solutions. Although an improvement was made to the Debye–Hückel equation in 1926 by Lars Onsager, also in 1923, Debye developed a theory to explain the Compton effect, the shifting of the frequency of X-rays when they interact with electrons. From 1934 to 1939 Debye was director of the section of the prestigious Kaiser Wilhelm Institute in Berlin. From 1936 onwards he was professor of Theoretical Physics at the Frederick William University of Berlin. These positions were held during the years that Adolf Hitler ruled Nazi Germany and, from 1938 onward, in 1939 Debye traveled to the United States to deliver the Baker Lectures at Cornell University in Ithaca, New York. After leaving Germany in early 1940, Debye became a professor at Cornell, chaired the department for 10 years. In 1946 he became an American citizen, unlike the European phase of his life, where he moved from city to city every few years, in the United States Debye remained at Cornell for the remainder of his career
50.
Paul Ehrenfest
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Paul Ehrenfest was born and grew up in Vienna in a Jewish family from Loštice in Moravia. His parents, Sigmund Ehrenfest and Johanna Jellinek, ran a grocery store, although the family was not overly religious, Paul studied Hebrew and the history of the Jewish people. Later he always emphasized his Jewish roots, Ehrenfest excelled in grade school but did not do well at the Akademisches Gymnasium, his best subject being mathematics. After transferring to the Franz Josef Gymnasium, his marks improved, in 1899 he passed the final exams. He majored in chemistry at the Institute of technology, but took courses at the University of Vienna, there he met his future wife Tatyana Afanasyeva, a young mathematician born in Kiev, then capital of the Kiev Governorate, Russian Empire, and educated in St Petersburg. In the spring of 1903 he met H. A, Lorentz during a short trip to Leiden. In the meantime he prepared a dissertation on Die Bewegung starrer Körper in Flüssigkeiten und die Mechanik von Hertz and he obtained his Ph. D. degree on June 23,1904 in Vienna, where he stayed from 1904 to 1905. On December 21,1904 he married Russian mathematician Tatyana Alexeyevna Afanasyeva and they had two daughters and two sons, Tatyana, also became a mathematician, Galinka, became an author and illustrator of childrens books, Paul, Jr. who also became a physicist, and Vassily. The Ehrenfests returned to Göttingen in September 1906 and they would not see Boltzmann again, on September 6 Boltzmann took his own life in Duino near Trieste. Ehrenfest published an obituary in which Boltzmanns accomplishments are described. Felix Klein, dean of the Göttinger mathematicians and chief editor of the Enzyklopädie der mathematischen Wissenschaften, had counted on Boltzmann for a review about statistical mechanics, now he asked Ehrenfest to take on this task. Together with his wife, Ehrenfest worked on it for several years, in 1907 the couple moved to St Petersburg. Ehrenfest found good friends there, in particular A. F. Joffe, moreover, as an Austrian citizen and of Jewish origin, he had no prospect of a permanent position. Early in 1912 Ehrenfest set out on a tour of German-speaking universities in the hope of a position and he visited Berlin where he saw Max Planck, Leipzig where he saw his old friend Herglotz, Munich where he met Arnold Sommerfeld, then Zürich and Vienna. While in Prague he met Albert Einstein for the first time, Einstein recommended Ehrenfest to succeed him in his position in Prague, but that did not work out. This was due to the fact that Ehrenfest declared himself to be an atheist, Sommerfeld offered him a position in Munich, but Ehrenfest received a better offer, at the same time there was an unexpected turn of events. H. A. Lorentz resigned his position as professor at the University of Leiden, in October 1912 Ehrenfest arrived in Leiden, and December 4 he gave his inaugural lecture Zur Krise der Lichtaether-Hypothese. He remained in Leiden for the rest of his career, in order to stimulate interaction and exchange among physics students he organized a discussion group and a fraternity called De Leidsche Flesch
51.
Albert Einstein
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Albert Einstein was a German-born theoretical physicist. He developed the theory of relativity, one of the two pillars of modern physics, Einsteins work is also known for its influence on the philosophy of science. Einstein is best known in popular culture for his mass–energy equivalence formula E = mc2, near the beginning of his career, Einstein thought that Newtonian mechanics was no longer enough to reconcile the laws of classical mechanics with the laws of the electromagnetic field. This led him to develop his theory of relativity during his time at the Swiss Patent Office in Bern. Briefly before, he aquired the Swiss citizenship in 1901, which he kept for his whole life and he continued to deal with problems of statistical mechanics and quantum theory, which led to his explanations of particle theory and the motion of molecules. He also investigated the properties of light which laid the foundation of the photon theory of light. In 1917, Einstein applied the theory of relativity to model the large-scale structure of the universe. He was visiting the United States when Adolf Hitler came to power in 1933 and, being Jewish, did not go back to Germany and he settled in the United States, becoming an American citizen in 1940. This eventually led to what would become the Manhattan Project, Einstein supported defending the Allied forces, but generally denounced the idea of using the newly discovered nuclear fission as a weapon. Later, with the British philosopher Bertrand Russell, Einstein signed the Russell–Einstein Manifesto, Einstein was affiliated with the Institute for Advanced Study in Princeton, New Jersey, until his death in 1955. Einstein published more than 300 scientific papers along with over 150 non-scientific works, on 5 December 2014, universities and archives announced the release of Einsteins papers, comprising more than 30,000 unique documents. Einsteins intellectual achievements and originality have made the word Einstein synonymous with genius, Albert Einstein was born in Ulm, in the Kingdom of Württemberg in the German Empire, on 14 March 1879. His parents were Hermann Einstein, a salesman and engineer, the Einsteins were non-observant Ashkenazi Jews, and Albert attended a Catholic elementary school in Munich from the age of 5 for three years. At the age of 8, he was transferred to the Luitpold Gymnasium, the loss forced the sale of the Munich factory. In search of business, the Einstein family moved to Italy, first to Milan, when the family moved to Pavia, Einstein stayed in Munich to finish his studies at the Luitpold Gymnasium. His father intended for him to electrical engineering, but Einstein clashed with authorities and resented the schools regimen. He later wrote that the spirit of learning and creative thought was lost in strict rote learning, at the end of December 1894, he travelled to Italy to join his family in Pavia, convincing the school to let him go by using a doctors note. During his time in Italy he wrote an essay with the title On the Investigation of the State of the Ether in a Magnetic Field
52.
Hugh Everett III
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Hugh Everett III was an American physicist who first proposed the many-worlds interpretation of quantum physics, which he termed his relative state formulation. Discouraged by the scorn of other physicists for MWI, Everett ended his physics career after completing his Ph. D, afterwards, he developed the use of generalized Lagrange multipliers for operations research and applied this commercially as a defense analyst and a consultant. He was married to Nancy Everett née Gore and they had two children, Elizabeth Everett and Mark Oliver Everett, who became frontman of the musical band Eels. Born in 1930, Everett was born and raised in the Washington, Everetts parents separated when he was young. Initially raised by his mother, he was raised by his father and stepmother from the age of seven, Everett won a half scholarship to St Johns College, a private military high school in Washington DC. From there he moved to the nearby Catholic University of America to study engineering as an undergraduate. While there he read about Dianetics in Astounding Science Fiction, although he never exhibited any interest in Scientology, he did retain a distrust of conventional medicine throughout his life. During World War II his father was fighting in Europe as a lieutenant colonel on the general staff. After World War II, Everetts father was stationed in West Germany, father and son were both keen photographers and took hundreds of pictures of West Germany being rebuilt. Reflecting their technical interests, the pictures were almost devoid of people, Everett graduated from The Catholic University of America in 1953 in chemical engineering, although he had completed sufficient courses for a mathematics degree as well. Everett then received a National Science Foundation fellowship that allowed him to attend Princeton University for graduate studies and he started his studies at Princeton in the Mathematics Department working on the then-new field of game theory under Albert W. Tucker, but slowly drifted into physics. In 1953 he started taking his first physics courses, notably Introductory Quantum Mechanics with Robert Dicke, during 1954, he attended Methods of Mathematical Physics with Eugene Wigner, although he remained active with mathematics and presented a paper on military game theory in December. He passed his examinations in the spring of 1955, thereby gaining his Masters degree. In his third year at Princeton Everett moved into an apartment which he shared with three friends he had made during his first year, Hale Trotter, Harvey Arnold and Charles Misner, Arnold later described Everett as follows, He was smart in a very broad way. I mean, to go from chemical engineering to mathematics to physics and spending most of the buried in a science fiction book, I mean. It was during this time that he met Nancy Gore, who typed up his Wave Mechanics Without Probability paper, Everett married Nancy Gore, the next year. The long paper was retitled as The Theory of the Universal Wave Function. Wheeler himself had traveled to Copenhagen in May,1956 with the goal of getting a favorable reception for at least part of Everetts work, but in vain
53.
Vladimir Fock
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Vladimir Aleksandrovich Fock was a Soviet physicist, who did foundational work on quantum mechanics and quantum electrodynamics. He was born in St. Petersburg, Russia, in 1922 he graduated from Petrograd University, then continued postgraduate studies there. He became a professor there in 1932, in 1926 he derived the Klein–Gordon equation. He gave his name to Fock space, the Fock representation and Fock state and he made many subsequent scientific contributions, during the rest of his life. Fock made significant contributions to relativity theory, specifically for the many body problems. In Leningrad, Fock created a school in theoretical physics. He wrote the first textbook on quantum mechanics Foundations of quantum mechanics, historians of science, such as Loren Graham, see Fock as a representative and proponent of Einsteins theory of relativity within the Soviet world. At a time when most Marxist philosophers objected to relativity theory and he was a full member of the USSR Academy of Sciences and a member of the International Academy of Quantum Molecular Science. Fock space Fock matrix Mehler–Fock transform Graham, L, the reception of Einsteins ideas, Two examples from contrasting political cultures. In Holton, G. and Elkana, Y. Albert Einstein, Princeton, NJ, Princeton UP, pp. 107–136 Fock, V. A. The Theory of Space, Time and Gravitation
54.
Enrico Fermi
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Enrico Fermi was an Italian physicist, who created the worlds first nuclear reactor, the Chicago Pile-1. He has been called the architect of the age and the architect of the atomic bomb. He was one of the few physicists to excel both theoretically and experimentally and he made significant contributions to the development of quantum theory, nuclear and particle physics, and statistical mechanics. Fermis first major contribution was to statistical mechanics, today, particles that obey the exclusion principle are called fermions. Later Pauli postulated the existence of an invisible particle emitted along with an electron during beta decay. Fermi took up this idea, developing a model that incorporated the postulated particle and his theory, later referred to as Fermis interaction and still later as weak interaction, described one of the four fundamental forces of nature. Fermi left Italy in 1938 to escape new Italian Racial Laws that affected his Jewish wife Laura Capon and he emigrated to the United States where he worked on the Manhattan Project during World War II. Fermi led the team designed and built Chicago Pile-1, which went critical on 2 December 1942. He was on hand when the X-10 Graphite Reactor at Oak Ridge, Tennessee, went critical in 1943, at Los Alamos he headed F Division, part of which worked on Edward Tellers thermonuclear Super bomb. He was present at the Trinity test on 16 July 1945, after the war, Fermi served under J. Robert Oppenheimer on the General Advisory Committee, which advised the Atomic Energy Commission on nuclear matters and policy. Following the detonation of the first Soviet fission bomb in August 1949 and he was among the scientists who testified on Oppenheimers behalf at the 1954 hearing that resulted in the denial of the latters security clearance. Enrico Fermi was born in Rome, Italy, on 29 September 1901 and he was the third child of Alberto Fermi, a division head in the Ministry of Railways, and Ida de Gattis, an elementary school teacher. His only sister, Maria, was two years older than he was, and his brother Giulio was a year older, after the two boys were sent to a rural community to be wet nursed, Enrico rejoined his family in Rome when he was two and a half. Although he was baptised a Roman Catholic in accordance with his grandparents wishes, his family was not particularly religious, as a young boy he shared the same interests as his brother Giulio, building electric motors and playing with electrical and mechanical toys. Giulio died during the administration of an anesthetic for an operation on a throat abscess in 1915, one of Fermis first sources for his study of physics was a book he found at the local market at Campo de Fiori in Rome. Published in 1840, the 900-page Elementorum physicae mathematicae, was written in Latin by Jesuit Father Andrea Caraffa and it covered mathematics, classical mechanics, astronomy, optics, and acoustics, insofar as these disciplines were understood when the book was written. Fermis interest in physics was encouraged by his fathers colleague Adolfo Amidei, who gave him several books on physics and mathematics. Fermi graduated from school in July 1918 and, at Amideis urging
55.
Richard Feynman
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For his contributions to the development of quantum electrodynamics, Feynman, jointly with Julian Schwinger and Sinichirō Tomonaga, received the Nobel Prize in Physics in 1965. Feynman developed a widely used pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, during his lifetime, Feynman became one of the best-known scientists in the world. In a 1999 poll of 130 leading physicists worldwide by the British journal Physics World he was ranked as one of the ten greatest physicists of all time. Along with his work in physics, Feynman has been credited with pioneering the field of quantum computing. Tolman professorship in physics at the California Institute of Technology. They were not religious, and by his youth, Feynman described himself as an avowed atheist, like Albert Einstein and Edward Teller, Feynman was a late talker, and by his third birthday had yet to utter a single word. He retained a Brooklyn accent as an adult and that accent was thick enough to be perceived as an affectation or exaggeration – so much so that his good friends Wolfgang Pauli and Hans Bethe once commented that Feynman spoke like a bum. The young Feynman was heavily influenced by his father, who encouraged him to ask questions to challenge orthodox thinking, from his mother, he gained the sense of humor that he had throughout his life. As a child, he had a talent for engineering, maintained a laboratory in his home. When he was in school, he created a home burglar alarm system while his parents were out for the day running errands. When Richard was five years old, his mother gave birth to a brother, Henry Philips. Four years later, Richards sister Joan was born, and the moved to Far Rockaway. Though separated by nine years, Joan and Richard were close and their mother thought that women did not have the cranial capacity to comprehend such things. Despite their mothers disapproval of Joans desire to study astronomy, Richard encouraged his sister to explore the universe, Joan eventually became an astrophysicist specializing in interactions between the Earth and the solar wind. Feynman attended Far Rockaway High School, a school in Far Rockaway, Queens, upon starting high school, Feynman was quickly promoted into a higher math class. A high-school-administered IQ test estimated his IQ at 125—high, but merely respectable according to biographer James Gleick and his sister Joan did better, allowing her to claim that she was smarter. Years later he declined to join Mensa International, saying that his IQ was too low, physicist Steve Hsu stated of the test, I suspect that this test emphasized verbal, as opposed to mathematical, ability. Feynman received the highest score in the United States by a margin on the notoriously difficult Putnam mathematics competition exam
56.
Roy Glauber
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Roy Jay Glauber is an American theoretical physicist. He is the Mallinckrodt Professor of Physics at Harvard University and Adjunct Professor of Optical Sciences at the University of Arizona. In this work, published in 1963, he created a model for photodetection and explained the fundamental characteristics of different types of light and his theories are widely used in the field of quantum optics. He currently serves on the National Advisory Board of the Center for Arms Control and Non-Proliferation, Glauber was born in 1925 in New York City. He was a member of the 1941 graduating class of the Bronx High School of Science, after his sophomore year he was recruited to work on the Manhattan Project, where he was one of the youngest scientists at Los Alamos National Laboratory. His work involved calculating the mass for the atom bomb. After two years at Los Alamos, he returned to Harvard, receiving his bachelors degree in 1946, glaubers recent research has dealt with problems in a number of areas of quantum optics, a field which, broadly speaking, studies the quantum electrodynamical interactions of light and matter. Glauber has received honors for his research, including the Albert A. This Nobel Prize is considered controversial within the community since only Glauber was awarded the Prize. On 22 April 2008, Professor Glauber was awarded the Medalla de Oro del CSIC in a ceremony held in Madrid and he was elected a Foreign Member of the Royal Society in 1997. He missed the 2005 event, though, as he was being awarded his real Nobel Prize at the time, Glauber currently lives in Arlington, Massachusetts. Glauber has a son and a daughter, and five grandchildren
57.
Martin Gutzwiller
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Martin Charles Gutzwiller was a Swiss-American physicist, known for his work on field theory, quantum chaos, and complex systems. He spent most of his career at IBM Research, and was also a professor of physics at Yale University. Gutzwiller was born on October 12,1925 in the Swiss city of Basel and he completed a Diploma degree from ETH Zurich, where he studied quantum physics under Wolfgang Pauli. He then went to the University of Kansas and completed a Ph. D under Max Dresden and he also held temporary teaching appointments at Columbia University, ETH Zurich, Paris-Orsay, and Stockholm. He was Vice Chair for the Committee on Mathematical Physics, of the International Union of Pure and Applied Physics and he joined Yale University as adjunct professor in 1993, retaining the position until his retirement. He was also the first to investigate the relationship between classical and quantum mechanics in chaotic systems, in that context, he developed the Gutzwiller trace formula, the main result of periodic orbit theory, which gives a recipe for computing spectra from periodic orbits of a system. He is the author of the monograph on the subject, Chaos in Classical. Gutzwiller is also known for finding solutions to mathematical problems in field theory, wave propagation, crystal physics. Gutzwiller had an avid interest in the history of science and he eventually acquired a valuable collection of rare books on astronomy and mechanics. Shortly after his death, his collection was auctioned at Swann Galleries, the auction took place on April 3,2014 and raised a total of US$341,788. Gutzwiller 1990 Baeriswyl, Dionys, Berry, Michael, Vollhardt, Dieter
58.
Werner Heisenberg
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Werner Karl Heisenberg was a German theoretical physicist and one of the key pioneers of quantum mechanics. He published his work in 1925 in a breakthrough paper, in the subsequent series of papers with Max Born and Pascual Jordan, during the same year, this matrix formulation of quantum mechanics was substantially elaborated. In 1927 he published his uncertainty principle, upon which he built his philosophy, Heisenberg was awarded the Nobel Prize in Physics for 1932 for the creation of quantum mechanics. He was a principal scientist in the Nazi German nuclear weapon project during World War II and he travelled to occupied Copenhagen where he met and discussed the German project with Niels Bohr. Following World War II, he was appointed director of the Kaiser Wilhelm Institute for Physics and he was director of the institute until it was moved to Munich in 1958, when it was expanded and renamed the Max Planck Institute for Physics and Astrophysics. He studied physics and mathematics from 1920 to 1923 at the Ludwig-Maximilians-Universität München, at Munich, he studied under Arnold Sommerfeld and Wilhelm Wien. At Göttingen, he studied physics with Max Born and James Franck and he received his doctorate in 1923, at Munich under Sommerfeld. He completed his Habilitation in 1924, at Göttingen under Born, at the event, Bohr was a guest lecturer and gave a series of comprehensive lectures on quantum atomic physics. There, Heisenberg met Bohr for the first time, and it had a significant, Heisenbergs doctoral thesis, the topic of which was suggested by Sommerfeld, was on turbulence, the thesis discussed both the stability of laminar flow and the nature of turbulent flow. The problem of stability was investigated by the use of the Orr–Sommerfeld equation and he briefly returned to this topic after World War II. Heisenbergs paper on the anomalous Zeeman effect was accepted as his Habilitationsschrift under Max Born at Göttingen, in his youth he was a member and Scoutleader of the Neupfadfinder, a German Scout association and part of the German Youth Movement. In August 1923 Robert Honsell and Heisenberg organized a trip to Finland with a Scout group of this association from Munich, Heisenberg arrived at Munich in 1919 as a member of Freikorps to fight the Bavarian Soviet Republic established a year earlier. Five decades later he recalled those days as youthful fun, like playing cops and robbers and so on, from 1924 to 1927, Heisenberg was a Privatdozent at Göttingen. His seminal paper, Über quantentheoretischer Umdeutung was published in September 1925 and he returned to Göttingen and with Max Born and Pascual Jordan, over a period of about six months, developed the matrix mechanics formulation of quantum mechanics. On 1 May 1926, Heisenberg began his appointment as a university lecturer and it was in Copenhagen, in 1927, that Heisenberg developed his uncertainty principle, while working on the mathematical foundations of quantum mechanics. On 23 February, Heisenberg wrote a letter to fellow physicist Wolfgang Pauli, in his paper on the uncertainty principle, Heisenberg used the word Ungenauigkeit. In 1927, Heisenberg was appointed ordentlicher Professor of theoretical physics and head of the department of physics at the Universität Leipzig, in his first paper published from Leipzig, Heisenberg used the Pauli exclusion principle to solve the mystery of ferromagnetism. Slater, Edward Teller, John Hasbrouck van Vleck, Victor Frederick Weisskopf, Carl Friedrich von Weizsäcker, Gregor Wentzel, in early 1929, Heisenberg and Pauli submitted the first of two papers laying the foundation for relativistic quantum field theory
59.
David Hilbert
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David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th, Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis, Hilbert adopted and warmly defended Georg Cantors set theory and transfinite numbers. A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems set the course for much of the mathematical research of the 20th century. Hilbert and his students contributed significantly to establishing rigor and developed important tools used in mathematical physics. Hilbert is known as one of the founders of theory and mathematical logic. In late 1872, Hilbert entered the Friedrichskolleg Gymnasium, but, after a period, he transferred to. Upon graduation, in autumn 1880, Hilbert enrolled at the University of Königsberg, in early 1882, Hermann Minkowski, returned to Königsberg and entered the university. Hilbert knew his luck when he saw it, in spite of his fathers disapproval, he soon became friends with the shy, gifted Minkowski. In 1884, Adolf Hurwitz arrived from Göttingen as an Extraordinarius, Hilbert obtained his doctorate in 1885, with a dissertation, written under Ferdinand von Lindemann, titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen. Hilbert remained at the University of Königsberg as a Privatdozent from 1886 to 1895, in 1895, as a result of intervention on his behalf by Felix Klein, he obtained the position of Professor of Mathematics at the University of Göttingen. During the Klein and Hilbert years, Göttingen became the preeminent institution in the mathematical world and he remained there for the rest of his life. Among Hilberts students were Hermann Weyl, chess champion Emanuel Lasker, Ernst Zermelo, john von Neumann was his assistant. At the University of Göttingen, Hilbert was surrounded by a circle of some of the most important mathematicians of the 20th century, such as Emmy Noether. Between 1902 and 1939 Hilbert was editor of the Mathematische Annalen, good, he did not have enough imagination to become a mathematician. Hilbert lived to see the Nazis purge many of the prominent faculty members at University of Göttingen in 1933 and those forced out included Hermann Weyl, Emmy Noether and Edmund Landau. One who had to leave Germany, Paul Bernays, had collaborated with Hilbert in mathematical logic and this was a sequel to the Hilbert-Ackermann book Principles of Mathematical Logic from 1928. Hermann Weyls successor was Helmut Hasse, about a year later, Hilbert attended a banquet and was seated next to the new Minister of Education, Bernhard Rust
60.
Pascual Jordan
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Ernst Pascual Jordan was a theoretical and mathematical physicist who made significant contributions to quantum mechanics and quantum field theory. He contributed much to the form of matrix mechanics. While the Jordan algebra is employed for and is used in studying the mathematical and conceptual foundations of quantum theory. An ancestor of Pascual Jordan named Pascual Jorda was a Spanish nobleman and cavalry officer who served with the British during, Jorda eventually settled in Hanover, which in those days was a possession of the British royal family. The family name was changed to Jordan. A family tradition dictated that the son in each generation be named Pascual. Jordan enrolled in the Hanover Technical University in 1921 where he studied an eclectic mix of zoology, mathematics, as was typical for a German university student of the time, he shifted his studies to another university before obtaining a degree. Göttingen University, his destination in 1923, was then at the zenith of its prowess and fame in mathematics. At Göttingen Jordan became an assistant first to mathematician Richard Courant, together with Max Born and Werner Heisenberg, Jordan was co-author of an important series of papers on quantum mechanics. He went on to early quantum field theory before largely switching his focus to cosmology before World War II. Jordan devised a type of non-associative algebras, now named Jordan algebras in his honor, in an attempt to create an algebra of observables for quantum mechanics, today, von Neumann algebras are also employed for this purpose. In 1966, Jordan published the 182 page work Die Expansion der Erde, the continents having to adapt to the ever flatter surface of the growing ball, the mountain ranges on the Earths surface would, in the course of that, have come into being as constricted folds. Despite the energy Jordan invested in the expanding Earth theory, his work was never taken seriously by either physicists or geologists. In 1933, Jordan joined the Nazi party, like Philipp Lenard and Johannes Stark, and, moreover, but at the same time, he remained a defender of Einstein and other Jewish scientists. Jordan enlisted in the Luftwaffe in 1939 and worked as a weather analyst at the Peenemünde rocket center, during the war he attempted to interest the Nazi party in various schemes for advanced weapons. His suggestions were ignored because he was considered unreliable, probably because of his past associations with Jews. Had Jordan not joined the Nazi party, it is conceivable that he could have won a Nobel Prize in Physics for his work with Max Born, Born would go on to win the 1954 Physics Prize with Walther Bothe. Jordan went against Paulis advice, and reentered politics after the period of came to an end under the pressures of the Cold War
61.
Hans Kramers
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Hendrik Anthony Hans Kramers was a Dutch physicist who worked with Niels Bohr to understand how electromagnetic waves interact with matter. Hans Kramers was born in Rotterdam, the son of Hendrik Kramers, a physician, and Jeanne Susanne Breukelman. In 1912 Hans finished secondary education in Rotterdam, and studied mathematics and physics at the University of Leiden, where he obtained a masters degree in 1916. Kramers wanted to obtain foreign experience during his research, but his first choice of supervisor. Because Denmark was neutral in war, as was the Netherlands, he travelled to Copenhagen. Bohr took him on as a Ph. D. candidate, although Kramers did most of his doctoral research in Copenhagen, he obtained his formal Ph. D. under Ehrenfest in Leiden, on 8 May 1919. Kramers greatly enjoyed music and could play the cello and the piano, after working for almost ten years in Bohrs group and becoming an associate professor at the University of Copenhagen, Kramers left Denmark in 1926 and returned to the Netherlands. He became a professor in theoretical physics at Utrecht University. In 1934 he left Utrecht and succeeded Paul Ehrenfest in Leiden, from 1931 until his death he held also a cross appointment at Delft University of Technology. Kramers was one of the founders of the Mathematisch Centrum in Amsterdam, in 1925, with Werner Heisenberg he developed the Kramers–Heisenberg dispersion formula. He is also credited with introducing in 1948 the concept of renormalization into quantum field theory, on 25 October 1920 he was married to Anna Petersen. They had three daughters and one son, Kramers became member of the Royal Netherlands Academy of Arts and Sciences in 1929, he was forced to resign in 1942. He joined the Academy again in 1945, Kramers won the Lorentz Medal in 1947 and Hughes Medal in 1951. Kramers – Between Tradition and Revolution, belinfante, F. J. ter Haar, D. Hendrik Anthony Kramers, 1894–1952. Casimir, Kramers, Hendrik Anthony, in Biografisch Woordenboek van Nederland, J. M. Romein, Hendrik Anthony Kramers, in, Jaarboek van de Maatschappij der Nederlandse Letterkunde te Leiden, 1951–1953, pp. 83–91
62.
Wolfgang Pauli
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Wolfgang Ernst Pauli was an Austrian-born Swiss and American theoretical physicist and one of the pioneers of quantum physics. The discovery involved spin theory, which is the basis of a theory of the structure of matter, Pauli was born in Vienna to a chemist Wolfgang Joseph Pauli and his wife Bertha Camilla Schütz, his sister was Hertha Pauli, the writer and actress. Paulis middle name was given in honor of his godfather, physicist Ernst Mach, Paulis paternal grandparents were from prominent Jewish families of Prague, his great-grandfather was the Jewish publisher Wolf Pascheles. Paulis father converted from Judaism to Roman Catholicism shortly before his marriage in 1899, Paulis mother, Bertha Schütz, was raised in her own mothers Roman Catholic religion, her father was Jewish writer Friedrich Schütz. Pauli was raised as a Roman Catholic, although eventually he and he is considered to have been a deist and a mystic. Pauli attended the Döblinger-Gymnasium in Vienna, graduating with distinction in 1918, only two months after graduation, he published his first paper, on Albert Einsteins theory of general relativity. He attended the Ludwig-Maximilians University in Munich, working under Arnold Sommerfeld, Sommerfeld asked Pauli to review the theory of relativity for the Encyklopädie der mathematischen Wissenschaften. Two months after receiving his doctorate, Pauli completed the article and it was praised by Einstein, published as a monograph, it remains a standard reference on the subject to this day. From 1923 to 1928, he was a lecturer at the University of Hamburg, during this period, Pauli was instrumental in the development of the modern theory of quantum mechanics. In particular, he formulated the principle and the theory of nonrelativistic spin. In 1928, he was appointed Professor of Theoretical Physics at ETH Zurich in Switzerland where he made significant scientific progress and he held visiting professorships at the University of Michigan in 1931, and the Institute for Advanced Study in Princeton in 1935. He was awarded the Lorentz Medal in 1931, at the end of 1930, shortly after his postulation of the neutrino and immediately following his divorce and the suicide of his mother, Pauli experienced a personal crisis. He consulted psychiatrist and psychotherapist Carl Jung who, like Pauli, Jung immediately began interpreting Paulis deeply archetypal dreams, and Pauli became one of the depth psychologists best students. He soon began to criticize the epistemology of Jungs theory scientifically, a great many of these discussions are documented in the Pauli/Jung letters, today published as Atom and Archetype. Jungs elaborate analysis of more than 400 of Paulis dreams is documented in Psychology, the German annexation of Austria in 1938 made him a German citizen, which became a problem for him in 1939 after the outbreak of World War II. In 1940, he tried in vain to obtain Swiss citizenship, Pauli moved to the United States in 1940, where he was employed as a professor of theoretical physics at the Institute for Advanced Study. In 1946, after the war, he became a citizen of the United States and subsequently returned to Zurich. In 1949, he was granted Swiss citizenship, in 1958, Pauli was awarded the Max Planck medal
63.
Willis Lamb
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Willis Eugene Lamb Jr. was an American physicist who won the Nobel Prize in Physics in 1955 for his discoveries concerning the fine structure of the hydrogen spectrum. The Nobel Committee that year awarded half the prize to Lamb and the half to Polykarp Kusch. Lamb was able to determine precisely a surprising shift in electron energies in a hydrogen atom, Lamb was a professor at the University of Arizona College of Optical Sciences. Lamb was born in Los Angeles, California, United States, first admitted in 1930, he received a Bachelor of Science in Chemistry from the University of California, Berkeley in 1934. For theoretical work on scattering of neutrons by a crystal, guided by J. Robert Oppenheimer, because of limited computational methods available at the time, this research narrowly missed revealing the Mössbauer Effect,19 years before its recognition by Mössbauer. He worked on theory, laser physics, and verifying quantum mechanics. Lamb was the Wykeham Professor of Physics at the University of Oxford from 1956 to 1962, and also taught at Yale, Columbia, Stanford and he was elected a Fellow of the American Academy of Arts and Sciences in 1963. Lamb is remembered as a rare theorist turned experimentalist by D. Kaiser, in addition to his crucial and famous contribution to quantum electrodynamics via the Lamb shift, in the latter part of his career he paid increasing attention to the field of quantum measurements. In one of his writings Lamb stated that most people who use quantum mechanics have little need to know much about the interpretation of the subject, Lamb was also openly critical of many of the interpretational trends on quantum mechanics. In 1939 Lamb married his first wife, Ursula Schaefer, a German student, after her death in 1996 he married physicist Bruria Kaufman in 1996, whom he later divorced. In 2008 he married Elsie Wattson, Lamb died on May 15,2008, at the age of 94, due to complications of a gallstone disorder. Biography and Bibliographic Resources, from the Office of Scientific and Technical Information Obituary, hans Bethe talking about Willis Lamb Willis E Lamb Award for Laser Science and Quantum Optics. Nobel lecture Collection of articles and group photograph, Obituary, Willis E. Lamb Jr.94, Nobel Prize-Winning Physicist National Academy of Sciences Biographical Memoir
64.
Lev Landau
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Lev Davidovich Landau was a Soviet physicist who made fundamental contributions to many areas of theoretical physics. Landaus father was an engineer with the oil industry and his mother was a doctor. He learned to differentiate at age 12 and to integrate at age 13, Landau graduated in 1920 at age 13 from gymnasium. His parents considered him too young to attend university, so for a year he attended the Baku Economical Technical School. In 1922, at age 14, he matriculated at the Baku State University, subsequently, he ceased studying chemistry, but remained interested in the field throughout his life. In 1924, he moved to the centre of Soviet physics at the time. In Leningrad, he first made the acquaintance of theoretical physics and dedicated himself fully to its study, Landau subsequently enrolled for post-graduate studies at the Leningrad Physico-Technical Institute where he eventually received a doctorate in Physical and Mathematical Sciences in 1934. By that time he was fluent in German and French and could communicate in English and he later improved his English and learned Danish. After brief stays in Göttingen and Leipzig, he went to Copenhagen on 8 April 1930 to work at the Niels Bohrs Institute for Theoretical Physics and he stayed there till 3 May of the same year. After the visit, Landau always considered himself a pupil of Niels Bohr, after his stay in Copenhagen, he visited Cambridge, where he worked with P. A. M. Dirac, Copenhagen, and Zurich, where he worked with Wolfgang Pauli. From Zurich Landau went back to Copenhagen for the third time, apart from his theoretical accomplishments, Landau was the principal founder of a great tradition of theoretical physics in Kharkov, Soviet Union, sometimes referred to as the Landau school. During the Great Purge, Landau was investigated within the UPTI Affair in Kharkov, Landau developed a famous comprehensive exam called the Theoretical Minimum which students were expected to pass before admission to the school. The exam covered all aspects of physics, and between 1934 and 1961 only 43 candidates passed, but those who did later became quite notable theoretical physicists. In 1932, he computed the Chandrashekhar limit, however, he did not apply it to white dwarf stars, Landau was the head of the Theoretical Division at the Institute for Physical Problems from 1937 until 1962. After his release Landau discovered how to explain Kapitsas superfluidity using sound waves, or phonons, Landau led a team of mathematicians supporting Soviet atomic and hydrogen bomb development. Landau calculated the dynamics of the first Soviet thermonuclear bomb, including predicting the yield, for this work he received the Stalin Prize in 1949 and 1953, and was awarded the title Hero of Socialist Labour in 1954. His students included Lev Pitaevskii, Alexei Abrikosov, Evgeny Lifshitz, Lev Gorkov, Isaak Khalatnikov, Roald Sagdeev and Isaak Pomeranchuk. He received the 1962 Nobel Prize in Physics for his development of a theory of superfluidity that accounts for the properties of liquid helium II at a temperature below 2.17 K